Cauchy's integral theorem and Green's Theorem clarification

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I was reading this page on Wikipedia: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem

In the very end, it says that "we therefore find that both integrands (and hence their integrals) are zero". Why? Shouldn't it be a constant?

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The author states that it is due to the Cauchy-Riemann equations that the integrands are zero.

And if the integrands are zero, then so are their corresponding integrals.