Math & Physics Problems Wikia
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Problem

If is analytic inside and on the boundary of a simply-connected region, prove the Cauchy integral formula

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CIF proof.gif

Solution

The function is analytic inside and on the boundary except at the point . Consider a circle of radius that is centered at . The equation of the circle is or in polar form where . Thus and . Consequently,

.

Taking the limit shows that

.

Therefore,

and

.
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