Poisson S Formula Let F Z Analytic Function Within Circle C Radius Prove Z Re R Polar Coo Q29624462

Poisson’s formula a) Let f(z) be an analytic function within and on the circle C of radius a. Prove that if z’ -re*, r < a, in polar coordinates located at the center of C. b) From the above formula, derive Poisson’s formula: 2? Jo a2 + r2-2ar cos (9-9) c) From Poisson’s formula, prove that if a function is analytic throughout and on a circle, then the value of the function at the center of the circle, f(0), is the average of its boundary values on the circle (mean value theorem) d) Now, starting with the equation 2 which differs from the equation of part (a) by the + sign, but holds for the same reason], derive [using part (c) eventually] e) Letting ??’) -/(reip) u(r, ?) +iv (r, ?), etc., deduce from (d) the formulas 2? 2? Jo 2? Jo These formulas express the real part of an analytic function on a circle in terms of its imaginary part on the circle, and vice versa. Many further applications of these formulas are possible. Show transcribed image text