I have the following problem [From: E. B. Saff & A. D. Snider: Fundamentals of Complex Analysis -- with Applications to Engineering and Science, pp. 375-376]:(adsbygoogle = window.adsbygoogle || []).push({});

Consider the problem of finding a function [itex]\phi[/itex] that is harmonic in the right half-plane and takes the values [itex]\phi(0,y)=y/\left(1+y^2\right)[/itex] on the imaginary axis.

According to the text the mappings (7)* and (8)** provide a correspondence between the right half-plane and the unit disk. (Of course, one should interchange the roles of z and w in the formulas). Thus the w-plane inherits from [itex]\phi(z)[/itex] a function [itex]\psi(w)[/itex] harmonic in the unit disk. Show that the values of [itex]\psi(w)[/itex] on the unit circle [itex]w=e^{i\theta}[/itex] must be given by

[tex]

\psi\left(e^{i\theta}\right)=\frac{\sin\theta}{2}~~(1)

[/tex]

*[itex]w=f(z)=\frac{1+z}{1-z}[/itex].

**[itex]z=\frac{w-1}{w+1}[/itex].

I know that (*) maps the unit circle onto the the imaginary axis and its interior onto the right half-plane. Furthermore, (**) maps the imaginary axis onto the unit circle, because (**) is the inverse of (*).

Now I have to find a function [itex]\psi(w)[/itex], whose values on the unit circle are given by (1). I am stuck here. I know that I have to use the definition of [itex]\psi[/itex]: [itex]\psi=\phi\circ f^{-1}[/itex], but I am not sure how to apply it for the actual problem.

Hopefully some of you could give me some hints. I do not ask for a solution to the problem, because that will not help me in future problems of this problem. The important thing is to understand the principle behind the solution procedure.

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# Homework Help: Dirichlet problem

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