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The following is an exercise from Complex Analysis by Stephen Fisher.

Fix a complex number $a$ and a positive real number $R$. Suppose $u$ is a function defined on the circle of radius $R$ centered at $a$. Let $C$ denote this circle.

Show that the average value of $u$ on $C$ is given by $\frac{1}{2\pi}\int_{0}^{2\pi} u(a + Re^{it})dt$.

Any Hints please.

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    What is the definition of the average value? (The integral is usually used as the definition, but I don't know the book you are using.)2012-10-26
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    The exercise is probably meant to relate to Cauchy's integral formula, but as said above: the integral you write is somewhat the definition of "average value" and the goal of the calculation, so the exercise is a bit unclear in its current form. I have the book (2nd edition at least) - which exercise number/page is it, so I can get a context?2012-10-26

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