Open mathieuboudreau opened 1 month ago
agahkarakuzu
commented
1 month ago @mathieuboudreau I'll need to try building it on my end, only then I can try to guess the source of this problem.
Figures/T1/foreword.md Cannot find image "intfig1.png" in Figures/T1Is this really the case?
mathieuboudreau
commented
1 month ago I removed it - it was an accidental copy-paste of an *.md file in its Figures folder
mathieuboudreau
commented
1 month ago I tried cleaning up some of the other warnings, here's the latest log
mathieuboudreau@Mathieus-MacBook-Pro mooc % myst build --pdf --force π¬ Performing exports: myst.yml -> _build/exports/myst.pdf π Built 1 Introduction to qMRI/01-intro.md in 6.99 s. π Built 1 Introduction to qMRI/02-million_dollar.md in 6.99 s. π Built 1 Introduction to qMRI/03-journey.md in 6.99 s. π Built 1 Introduction to qMRI/05-sequences.md in 6.99 s. π Built 1 Introduction to qMRI/06-qmri_practice.md in 6.99 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/01-introduction.md in 6.99 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/02-IR_SignalModelling.md in 6.99 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md in 6.99 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/04-IR_BenefitsAndPitfalls.md in 6.99 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/05-IR_OtherMethods.md in 6.99 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/01-Introduction.md in 6.99 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/02-VFA_SignalModelling.md in 6.99 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/03-VFA_DataFitting.md in 6.99 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/04-VFA_BenefitsAndPitfalls.md in 6.99 s. π Built 2 T1 Mapping/2-3 MP2RAGE/01-Introduction.md in 6.99 s. π Built 2 T1 Mapping/2-3 MP2RAGE/02-SignalModelling.md in 6.99 s. π Built 2 T1 Mapping/2-3 MP2RAGE/03-DataFitting.md in 6.98 s. π Built 2 T1 Mapping/2-3 MP2RAGE/04-BenefitsAndPitfalls.md in 6.98 s. π Built 2 T1 Mapping/2-4 Exercises/01-Exercises.md in 6.98 s. π Built 3 T2 Mapping/01-Introduction/01-Introduction.md in 6.98 s. π Built 3 T2 Mapping/01-Introduction/02-T2 mapping vs T2-weighted imaging.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/01-Introduction.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/02-Signal modelling.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/00-Foreward.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/01-T2.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/02-Noise.md in 6.98 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/04-Benefits and pitfalls.md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/01-Introduction.md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/02-SignalModelling.md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/03-DataFitting.md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/01-Myelin water fraction (MWF) imaging.md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/02-T2 and quantitative susceptibility mapping (QSM).md in 6.98 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/03-Benefits and pitfalls of multi-exponential T2 mapping.md in 6.98 s. π Built 4 B1 Mapping/01-Double Angle technique/01-Transmit and Receive RF field amplitudes.md in 6.98 s. π Built 4 B1 Mapping/01-Double Angle technique/02-Double Angle method(s).md in 6.98 s. π Built 4 B1 Mapping/01-Double Angle technique/03- Gradient echo and 180 deg spin echo method copy.md in 6.98 s. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. π Built 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md in 6.99 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/01-abstract.md in 6.99 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/02-SignalModeling.md in 6.99 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/03-DataFitting.md in 6.99 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/04-BenefitsAndPitfalls.md in 6.99 s. π Built 4 B1 Mapping/03-Filtering/01-Introduction.md in 6.99 s. π Built 4 B1 Mapping/03-Filtering/02-Filters and smoothing.md in 6.99 s. π Built 4 B1 Mapping/03-Filtering/03-1D examples.md in 6.99 s. π Built 4 B1 Mapping/03-Filtering/04-B1 map examples.md in 6.99 s. π Built 4 B1 Mapping/03-Filtering/05-Recommendations, benefits, and pitfalls.md in 6.99 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/1-Introduction.md in 6.99 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/2-Sources/01-Hardware.md in 6.99 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/2-Sources/02-Magnetic susceptibility.md in 6.99 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/3-Effects on signal.md in 6.99 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/01-Introduction.md in 6.99 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/02-Signal Theory.md in 6.99 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/03-Single Frequency Population.md in 6.99 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/04-Multi-Frequency Population.md in 6.99 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/05-Benefits and Pitfalls.md in 6.99 s. π Built 5 B0 Mapping/3 Phase Unwrapping/01-Introduction.md in 6.98 s. π Built 5 B0 Mapping/3 Phase Unwrapping/02-Temporal Unwrapping.md in 6.98 s. π Built 5 B0 Mapping/3 Phase Unwrapping/03-Spatial Unwrapping.md in 6.98 s. π Built 5 B0 Mapping/3 Phase Unwrapping/04-Phase unwrapping ambiguity.md in 6.98 s. π Built 5 B0 Mapping/3 Phase Unwrapping/05-Problematic phase map properties.md in 6.98 s. π Built 5 B0 Mapping/3 Phase Unwrapping/06-Software.md in 6.98 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/01-Introduction.md in 6.98 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/02-MultiEcho.md in 6.98 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/03-Reducing eddy currents.md in 6.98 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/04-Realtime.md in 6.98 s. π Built 5 B0 Mapping/5 Conclusion/5 Conclusion.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/01-abstract.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/02-SignalModelling.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/03-DataFitting.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/04-Summary.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/01-Introduction.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/02-MTR in theory.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/03-MTR in practice.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/04-Parting Thoughts.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/A1- Appendix A.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/01-Introduction.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/02-Theory.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/03-Simulations.md in 6.98 s. β οΈ 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \ S{0}=0.087\frac{1.8\cdot 0.03}{\frac{0.087^{2}}{2}+0+1.8\cdot 0.03}A β οΈ 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode S{MT}=0.044A\text{, (refactor)} \ π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md in 6.98 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A2-Appendix B.md in 6.98 s. π Built index.md in 6.66 s. πͺ Linked 1 DOI from doi.org in 6.08 s for 1 Introduction to qMRI/04-encode.md π Built 1 Introduction to qMRI/04-encode.md in 7.04 s. π Built Figures/T1/figure_2-2.ipynb in 7 s. π Built Figures/T1/figure_2-3.ipynb in 7 s. π Built Figures/T1/figure_2-4.ipynb in 6.99 s. π Built 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Unhandled LaTeX conversion for node of "bibliography" β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'fid' β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'adc' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mrm' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'mrm' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'fid' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'sar' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'te' βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'mri' βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'despot1' π Exported TeX in 6.16 s, copying to _build/temp/mystdpEhTq/myst.tex π¨ Rendering PDF (default) to _build/temp/myst5YYFpa/myst.pdf Error: ENOENT: no such file or directory, open '/Users/mathieuboudreau/neuropoly/mooc/repos/mooc/_build/temp/myst5YYFpa/myst.log' mathieuboudreau@Mathieus-MacBook-Pro mooc %
mathieuboudreau
commented
1 month ago Unhandled LaTeX conversion for node of "exercise"
These are annoying, I imaging that there is some "command" in those files that aren't actual latex and it doesn't know how to interpret them?
agahkarakuzu
commented
1 month ago OK, it was failing because date and authors were missing from myst.yml. That's when exports are defined globally in myst.yml.
Then I got this log:
It builds a 15MB broken pdf.
mathieuboudreau
commented
1 month ago in the "log" file (which is misnamed, maybe that's why it can't find it) that is mentioned here https://github.com/jupyter-book/mystmd/issues/1459, I get this
/bin/sh: latexmk: command not found
Tried installing latexmk but got frustrated - shouldn't it be packaged with myst?
agahkarakuzu
commented
1 month ago There are external dependencies for certain myst functionalities, Latex is one of them:
https://mystmd.org/guide/installing#dependencies-for-latex-and-pdf
Imagemagick, webp, etc are some other.
mathieuboudreau
commented
1 month ago It builds a 15MB broken pdf.
I guess MyST only promised that MyST projects can be compiled to PDF, maybe in the fine print they don't guarantee a readable PDF? lol
mathieuboudreau
commented
1 month ago There are external dependencies for certain myst functionalities, Latex is one of them:
https://mystmd.org/guide/installing#dependencies-for-latex-and-pdf
Imagemagick, webp, etc are some other.
I thought that might have been the case - I had checked earlier but must not have scrolled down far enough, missed it. Thanks! I'll try again after all that (hopefully) installs
agahkarakuzu
commented
1 month ago OK I've found it! There are some leftover {bibliography} directives from JupyterBook you'll need to remove them.
Also:
date and authors to myst.yml when using export with -pdf set globally. It will still give an error (Unhandled LaTeX conversion for node of "details", not sure where it is coming from), but should build a pdf that you can open :)
mathieuboudreau
commented
1 month ago I've tried installing the necessary latex dependencies (tlmgr, latexmk) but every steps seems to break. I got tlmgr installed, but then got these errors:
mooc_venv) mathieuboudreau@Mathieus-MacBook-Pro mooc % sudo tlmgr install latexmk
Password: Can't locate TeXLive/TLConfig.pm in @INC (you may need to install the TeXLive::TLConfig module) (@INC contains: /Users/mathieuboudreau/opt/anaconda3/envs/mooc_venv/share/texmf-dist/scripts/texlive /Users/mathieuboudreau/opt/anaconda3/envs/mooc_venv/share/tlpkg /Library/Perl/5.34/darwin-thread-multi-2level /Library/Perl/5.34 /Network/Library/Perl/5.34/darwin-thread-multi-2level /Network/Library/Perl/5.34 /Library/Perl/Updates/5.34.1 /System/Library/Perl/5.34/darwin-thread-multi-2level /System/Library/Perl/5.34 /System/Library/Perl/Extras/5.34/darwin-thread-multi-2level /System/Library/Perl/Extras/5.34) at /Users/mathieuboudreau/opt/anaconda3/envs/mooc_venv/bin/tlmgr line 92. BEGIN failed--compilation aborted at /Users/mathieuboudreau/opt/anaconda3/envs/mooc_venv/bin/tlmgr line 92. (mooc_venv) mathieuboudreau@Mathieus-MacBook-Pro mooc % sudo tlmgr install l
And online it seems the fix is pretty messy (https://tex.stackexchange.com/questions/539545/cant-locate-texlive-tlconfig-pm-in-inc), so I don't care to try much further getting this MyST PDF build locally
@agahkarakuzu I'm having trouble building a PDF of this MyST project - both on my mac and laptop, it stops and says there should be a log file but it is non existent. Any clues on how to debug this / get a PDF?
mathieuboudreau@Mathieus-MacBook-Pro mooc % myst build --pdf
π¬ Performing exports: myst.yml -> _build/exports/myst.pdf No character metrics for 'ΙΈ' in style 'Main-Regular' and mode 'text' No character metrics for 'ΙΈ' in style 'Main-Regular' and mode 'text' No character metrics for 'ΙΈ' in style 'Main-Regular' and mode 'text' No character metrics for 'ΙΈ' in style 'Main-Regular' and mode 'text' No character metrics for 'ΙΈ' in style 'Main-Regular' and mode 'text' π Built 1 Introduction to qMRI/01-intro.md in 6.35 s. π Built 1 Introduction to qMRI/02-million_dollar.md in 6.35 s. π Built 1 Introduction to qMRI/03-journey.md in 6.35 s. π Built 1 Introduction to qMRI/05-sequences.md in 6.35 s. π Built 1 Introduction to qMRI/06-qmri_practice.md in 6.35 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/01-introduction.md in 6.34 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/02-IR_SignalModelling.md in 6.34 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md in 6.34 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/04-IR_BenefitsAndPitfalls.md in 6.34 s. π Built 2 T1 Mapping/2-1 Inversion Recovery/05-IR_OtherMethods.md in 6.34 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/01-Introduction.md in 6.34 s. β οΈ 2 T1 Mapping/2-2 Variable Flip Angle/02-VFASignalModelling.md unknownSymbol, Unrecognized Unicode character "ΙΈ" (632) ΙΈ{n} β οΈ 2 T1 Mapping/2-2 Variable Flip Angle/02-VFASignalModelling.md unknownSymbol, Unrecognized Unicode character "ΙΈ" (632) ΙΈ{n-1} β οΈ 2 T1 Mapping/2-2 Variable Flip Angle/02-VFASignalModelling.md unknownSymbol, Unrecognized Unicode character "ΙΈ" (632) ΙΈ{0} β οΈ 2 T1 Mapping/2-2 Variable Flip Angle/02-VFASignalModelling.md unknownSymbol, Unrecognized Unicode character "ΙΈ" (632) ΙΈ{0} β οΈ 2 T1 Mapping/2-2 Variable Flip Angle/02-VFASignalModelling.md unknownSymbol, Unrecognized Unicode character "ΙΈ" (632) ΙΈ{0} π Built 2 T1 Mapping/2-2 Variable Flip Angle/02-VFA_SignalModelling.md in 6.34 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/03-VFA_DataFitting.md in 6.35 s. π Built 2 T1 Mapping/2-2 Variable Flip Angle/04-VFA_BenefitsAndPitfalls.md in 6.35 s. π Built 2 T1 Mapping/2-3 MP2RAGE/01-Introduction.md in 6.35 s. π Built 2 T1 Mapping/2-3 MP2RAGE/02-SignalModelling.md in 6.35 s. π Built 2 T1 Mapping/2-3 MP2RAGE/03-DataFitting.md in 6.35 s. π Built 2 T1 Mapping/2-3 MP2RAGE/04-BenefitsAndPitfalls.md in 6.34 s. π Built 2 T1 Mapping/2-4 Exercises/01-Exercises.md in 6.34 s. π Built 3 T2 Mapping/01-Introduction/01-Introduction.md in 6.34 s. π Built 3 T2 Mapping/01-Introduction/02-T2 mapping vs T2-weighted imaging.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/01-Introduction.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/02-Signal modelling.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/00-Foreward.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/01-T2.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/03-Data Fitting/02-Noise.md in 6.34 s. π Built 3 T2 Mapping/02-Monoexponential T2 Mapping/04-Benefits and pitfalls.md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/01-Introduction.md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/02-SignalModelling.md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/03-DataFitting.md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/01-Myelin water fraction (MWF) imaging.md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/02-T2 and quantitative susceptibility mapping (QSM).md in 6.34 s. π Built 3 T2 Mapping/03-Multiexponential T2 Mapping/04-Applications/03-Benefits and pitfalls of multi-exponential T2 mapping.md in 6.34 s. π Built 4 B1 Mapping/01-Double Angle technique/01-Transmit and Receive RF field amplitudes.md in 6.34 s. π Built 4 B1 Mapping/01-Double Angle technique/02-Double Angle method(s).md in 6.34 s. π Built 4 B1 Mapping/01-Double Angle technique/03- Gradient echo and 180 deg spin echo method copy.md in 6.34 s. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. β οΈ 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \begin{split} \text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \ &= \text{e}^{i\alpha+i\alpha} \ &= \text{e}^{i\alpha}\text{e}^{i\alpha} \ &= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\ &= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\ \text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\ \end{split} \ \ \text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\ \text{if }z=q \ \text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \ \text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \ \text{thus,} \ \ \begin{split} \text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\ \text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \ \end{split}\ Q.E.D. π Built 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md in 6.35 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/01-abstract.md in 6.35 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/02-SignalModeling.md in 6.35 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/03-DataFitting.md in 6.35 s. π Built 4 B1 Mapping/02-Actual Flip Angle Imaging/04-BenefitsAndPitfalls.md in 6.35 s. π Built 4 B1 Mapping/03-Filtering/01-Introduction.md in 6.35 s. π Built 4 B1 Mapping/03-Filtering/02-Filters and smoothing.md in 6.35 s. π Built 4 B1 Mapping/03-Filtering/03-1D examples.md in 6.35 s. π Built 4 B1 Mapping/03-Filtering/04-B1 map examples.md in 6.35 s. π Built 4 B1 Mapping/03-Filtering/05-Recommendations, benefits, and pitfalls.md in 6.35 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/1-Introduction.md in 6.35 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/2-Sources/01-Hardware.md in 6.35 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/2-Sources/02-Magnetic susceptibility.md in 6.35 s. π Built 5 B0 Mapping/1 B0 Inhomogeneities/3-Effects on signal.md in 6.35 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/01-Introduction.md in 6.35 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/02-Signal Theory.md in 6.35 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/03-Single Frequency Population.md in 6.34 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/04-Multi-Frequency Population.md in 6.34 s. π Built 5 B0 Mapping/2 Dual echo B0 mapping/05-Benefits and Pitfalls.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/01-Introduction.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/02-Temporal Unwrapping.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/03-Spatial Unwrapping.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/04-Phase unwrapping ambiguity.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/05-Problematic phase map properties.md in 6.34 s. π Built 5 B0 Mapping/3 Phase Unwrapping/06-Software.md in 6.34 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/01-Introduction.md in 6.34 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/02-MultiEcho.md in 6.34 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/03-Reducing eddy currents.md in 6.34 s. π Built 5 B0 Mapping/4 Advanced B0 Mapping Methods/04-Realtime.md in 6.34 s. π Built 5 B0 Mapping/5 Conclusion/5 Conclusion.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/01-abstract.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/02-SignalModelling.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/03-DataFitting.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/1 Quantitative Magnetization Transfer/04-Summary.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/01-Introduction.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/02-MTR in theory.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/03-MTR in practice.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/04-Parting Thoughts.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/2 Magnetization Transfer Ratio/A1- Appendix A.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/01-Introduction.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/02-Theory.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/03-Simulations.md in 6.34 s. β οΈ 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode \ S{0}=0.087\frac{1.8\cdot 0.03}{\frac{0.087^{2}}{2}+0+1.8\cdot 0.03}A β οΈ 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md newLineInDisplayMode, In LaTeX, \ or \newline does nothing in display mode S{MT}=0.044A\text{, (refactor)} \ π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A1-Appendix A.md in 6.34 s. π Built 6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/A2-Appendix B.md in 6.34 s. π Built Figures/T1/foreword.md in 6.32 s. π Built index.md in 6.01 s. πͺ Linked 1 DOI from doi.org in 5.5 s for 1 Introduction to qMRI/04-encode.md π Built 1 Introduction to qMRI/04-encode.md in 6.4 s. π Built Figures/T1/figure_2-2.ipynb in 6.35 s. π Built Figures/T1/figure_2-3.ipynb in 6.35 s. π Built Figures/T1/figure_2-4.ipynb in 6.35 s. π Built Figures/T1/figure_2-5.ipynb in 6.35 s. π Built Figures/T1/figure_2-6.ipynb in 6.34 s. π Built Figures/T1/figure_vfa2.ipynb in 6.34 s. π Built Figures/T1/figure_vfa3.ipynb in 6.34 s. π Built Figures/T1/figure_vfa4.ipynb in 6.34 s. π Built Figures/T1/figure_vfa5.ipynb in 6.34 s. π Built Figures/T1/figure_vfa6.ipynb in 6.34 s. π Built Figures/T1/figure_vfa7.ipynb in 6.34 s. π Built Figures/T1/mp2rageFig1.ipynb in 6.33 s. π Built Figures/T1/mp2rageFig2.ipynb in 6.33 s. π Built Figures/T2/fig3p1.ipynb in 6.32 s. π Built Figures/T2/fig3p2.ipynb in 6.31 s. π Built Figures/T2/fig3p3.ipynb in 6.31 s. π Built Figures/T2/fig3p4.ipynb in 6.31 s. π Built Figures/T2/fig3p5.ipynb in 6.31 s. π Built Figures/B1/afiFig1.ipynb in 6.31 s. π Built Figures/B1/afiFig2.ipynb in 6.31 s. π Built Figures/B1/afiFig3.ipynb in 6.3 s. π Built Figures/B1/afiFig4.ipynb in 6.3 s. π Built Figures/B1/afiFig5.ipynb in 6.3 s. π Built Figures/B1/afiFig6.ipynb in 6.3 s. π Built Figures/B1/afiFig7.ipynb in 6.3 s. π Built Figures/B1/afiFig8.ipynb in 6.29 s. π Built Figures/B1/daFig1.ipynb in 6.29 s. π Built Figures/B1/daFig2.ipynb in 6.29 s. π Built Figures/B1/daFig3.ipynb in 6.29 s. π Built Figures/B1/filtFig1.ipynb in 6.29 s. π Built Figures/B1/filtFig2.ipynb in 6.28 s. π Built Figures/B1/filtFig3.ipynb in 6.28 s. π Built Figures/B1/filtFig4.ipynb in 6.27 s. π Built Figures/B1/filtFig5.ipynb in 6.27 s. π Built Figures/B1/filtFig6.ipynb in 6.25 s. π Built Figures/B1/filtFig7.ipynb in 6.25 s. π Built Figures/B1/filtFig8.ipynb in 6.25 s. π Built Figures/B1/filtFig9.ipynb in 6.24 s. π Built Figures/B0/fig5p1.ipynb in 6.22 s. π Built Figures/B0/fig5p2.ipynb in 6.22 s. π Built Figures/B0/fig5p3.ipynb in 6.22 s. π Built Figures/B0/fig5p4.ipynb in 6.17 s. π Built Figures/B0/fig5p5.ipynb in 6.16 s. π Built Figures/B0/fig5p6.ipynb in 6.12 s. π Built Figures/B0/fig5p7.ipynb in 6.11 s. π Built Figures/B0/fig5p8.ipynb in 6.1 s. π Built Figures/B0/fig5p9.ipynb in 6.1 s. π Built Figures/B0/fig5p10.ipynb in 6.09 s. π Built Figures/B0/fig5p11.ipynb in 6.09 s. π Built Figures/B0/fig5p12.ipynb in 6.09 s. π Built Figures/B0/fig5p13.ipynb in 6.09 s. π Built Figures/B0/fig5p14.ipynb in 6.07 s. π Built Figures/B0/fig5p15.ipynb in 6.05 s. π Built Figures/B0/fig5p16.ipynb in 6.05 s. π Built Figures/B0/fig5p17.ipynb in 6.05 s. π Built Figures/B0/fig5p18.ipynb in 6.05 s. π Built Figures/MT/mtrFig1.ipynb in 6.05 s. π Built Figures/MT/mtrFig2.ipynb in 6.05 s. π Built Figures/MT/mtrFig3.ipynb in 6.05 s. π Built Figures/MT/mtrFig4.ipynb in 6.05 s. π Built Figures/MT/mtrFig5.ipynb in 6.05 s. π Built Figures/MT/mtsatAppendixB1.ipynb in 6.05 s. π Built Figures/MT/mtsatAppendixB2.ipynb in 6.04 s. π Built Figures/MT/mtsatAppendixB3.ipynb in 6.04 s. π Built Figures/MT/mtsatFig1.ipynb in 6.04 s. π Built Figures/MT/mtsatFig2.ipynb in 6.04 s. π Built Figures/MT/mtsatFig3.ipynb in 6.04 s. π Built Figures/MT/qmtFig1.ipynb in 6.04 s. π Built Figures/MT/qmtFig2.ipynb in 6.04 s. π Built Figures/MT/qmtFig3.ipynb in 6.04 s. π Built Figures/MT/qmtFig4.ipynb in 6.04 s. π Built Figures/MT/qmtFig5.ipynb in 6.04 s. π Built Figures/MT/qmtFig6.ipynb in 6.04 s. π Built 156 pages for export (including 1 dependency) from /Users/mathieuboudreau/neuropoly/mooc/repos/mooc in 7.6 s. βοΈ Figures/T1/foreword.md Cannot find image "intfig1.png" in Figures/T1 β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'ti' βοΈ Unhandled LaTeX conversion for node of "bibliography" β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'ti' βοΈ Unhandled LaTeX conversion for node of "bibliography" β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'rd-nls' β οΈ myst.yml duplicate abbreviation definition for 'te' βοΈ Unhandled LaTeX conversion for node of "bibliography" β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ 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abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'fid' β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'adc' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mrm' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'mrm' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'fid' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'b0' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'gre' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'se' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'sar' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'rf' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'fa' β οΈ myst.yml duplicate abbreviation definition for 'te' βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" βοΈ Unhandled LaTeX conversion for node of "details" β οΈ myst.yml duplicate abbreviation definition for 'mtsat' β οΈ myst.yml duplicate abbreviation definition for 'mt' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'mtr' β οΈ myst.yml duplicate abbreviation definition for 'qmt' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'qmri' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 't2' β οΈ myst.yml duplicate abbreviation definition for 'nmr' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'te' β οΈ myst.yml duplicate abbreviation definition for 't1' β οΈ myst.yml duplicate abbreviation definition for 'tr' βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" βοΈ Unhandled LaTeX conversion for node of "exercise" β οΈ myst.yml duplicate abbreviation definition for 'csf' β οΈ myst.yml duplicate abbreviation definition for 'mri' β οΈ myst.yml duplicate abbreviation definition for 'tr' β οΈ myst.yml duplicate abbreviation definition for 'vfa' β οΈ myst.yml duplicate abbreviation definition for 'spgr' β οΈ myst.yml duplicate abbreviation definition for 'snr' β οΈ myst.yml duplicate abbreviation definition for 'ti' β οΈ myst.yml duplicate abbreviation definition for 'despot1' π Exported TeX in 5.81 s, copying to _build/temp/myst2IM4FJ/myst.tex π¨ Rendering PDF (default) to _build/temp/mystK49MDn/myst.pdf Error: ENOENT: no such file or directory, open '/Users/mathieuboudreau/neuropoly/mooc/repos/mooc/_build/temp/mystK49MDn/myst.log' mathieuboudreau@Mathieus-MacBook-Pro mooc %