For an quiz for a diff eq class, I need to prove the Cauchy integral formula.
The assignment says prove the formula for analytic functions. Is the proof significantly different when the function is not analytic?
Basically, a proof I found online says that I take the integral of a larger contour C that encloses the contour C_r which is the contour that wraps around the cirlce of radius r around z_0. By the deformation principle, if r is sufficiently small, then the integral of f(z) around C and C_r are equal. I've tried searching for 'deformation principle' but none was of much help.
What is the deformation principle? I don't know much about contour integrals, so if someone can relate it to just regular integrals, that'd be great.
http://www2.latech.edu/~schroder/slides/comp_var/14_Cauchy_Integral_Formula.pdf [Broken] Pg 43
What is max C(r, z_0) supposed to mean? I think the this proof uses the same principles as in the first one, but doesn't explicitly use the delta-epsilon limit.
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