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dlfivefifty / M3M6MethodsOfMathematicalPhysics

Lecture notes for M3M6 Methods of Mathematical Physics
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May be lost a minus in the equation? #2

Closed Jiaqi-knight closed 5 years ago

Jiaqi-knight commented 5 years ago

Lecture 9: Cauchy transforms

  1. Cauchy transforms on the interval
    • Plemelj's theorem
  2. Hilbert transform ...... are "nice", we can deform ΓΓ to be on the contour itself, giving
    $$\phi(z) = -{1 \over 2 \pi \I} \int_{-1}^1 {\phi^+(x) - \phi^-(x) \over x - z} \dx$$
Jiaqi-knight commented 5 years ago

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dlfivefifty commented 5 years ago

I don’t think so: the first equation corresponds to integrating clockwise (without the - sign). This is the same as integrating above the contour left-to-right then below the contour right-to-left.

Jiaqi-knight commented 5 years ago

Sorry, It's my fault!! I was confused with two contours, Thanks.

Jiaqi-knight commented 5 years ago

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