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sigmund
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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]:
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.
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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