New math, plus script to download latex images of equations

[DLu]

Extension of #37 with additional explanation of the equation/derivations. CC: @EwingKang

Formatted README

Formatted Derivation

[EwingKang]

So, to figure out where the numbers came from, I have tested 2 additional points.

• Historic numbers:
  C0=0.7040; C1=0.5307; C2=-0.2301

• Delta=0 (by @DLu ) C0=0.7071; C1=0.5; C2=-0.1768

• Delta = 1/2 C0=0.7019; C1=0.5270; C2=-0.2160

• Delta = 1 C0=0.875; C1=0.5; C2=-0.25 or C0=7/8; C1=1/2; C2=-1/4

[DLu]

I'm not sure where the difference is, but I'm not getting the same values as you for the coefficients. Can you check out this plot and see if it makes sense to you.

[EwingKang]

Hello, @DLu To my understanding in Taylor series expansion, you have to multiply the "difference between variable and expansion point" after the coefficient. This is also the case mentioned in the formatted derivation.

In the case of expansion at [d]=0, it is a special case that the coefficient of the nth-order [d] is actually nth-order derivative of the function (times 1/n!).

It could be I'm wrong somewhere, hope this math make sense to you:

[DLu]

@EwingKang - You're right...thanks for keeping me honest. I've updated the PR with the general equations for the different Taylor Series. The historical values are now pretty close to a=0.5. Also, what I've calculated for the coefficients matches what you have for delta=0.5, but not delta=1.0 (which is off by a factor of 2).

[EwingKang]

@DLu I've re-calculate delta=1 and you are right, it should be: C2= -0.5, C1=1, C0=1/2 as the new readme listed. I see no further issue regarding the math portion of the document. :smiley: Thank you (especially for your patient) !!!