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A full statement of Poisson’s integral formulas for a circle is as follows:

Theorem. Let f(z) be analytic inside and on the circle C defined by |z| = R, and let z = re be any point inside C. See Fig. 2 below. Then

If u(r, θ) and v u(r, θ) are the real and imaginary parts of f(re) while u(R, θ) and v(R, θ) are the real and imaginary parts of f(Re), then

Proof. The proof utilizes the following concept:

Def. Inverse of a point with respect to a circle. Let C be a circle with center at O and let P be any point inside or outside C. See Fig. 1. Draw line OA through point P. Then the inverse of point P is the point P' located on line OA whose distance from O is such that

Either of the points P and P' is called the inverse of the other and the center of the circle is called the center of inversion.

Since z = re is any point inside C, we have by Cauchy’s integral formula

The inverse of the point P with respect to C lies outside C and is given by R2/ . Thus by Cauchy’s theorem,

We now subtract 5) from 4) to obtain

Now let z = re and w = Re. Then since = re-iθ, we have

which is 1) above.

Since f(re) = u(r,θ) + i v(r,θ) and f(R ) = u(R, ) + i v(R, ) we obtain from 1)

Thus we have 2) and 3) above

Source: Spiegel. Complex Variables (Schaum). p. 129-130

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