# Dirichlet problem

1. Oct 10, 2005

### sigmund

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 $\phi$ that is harmonic in the right half-plane and takes the values $\phi(0,y)=y/\left(1+y^2\right)$ 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 $\phi(z)$ a function $\psi(w)$ harmonic in the unit disk. Show that the values of $\psi(w)$ on the unit circle $w=e^{i\theta}$ must be given by
$$\psi\left(e^{i\theta}\right)=\frac{\sin\theta}{2}~~(1)$$

*$w=f(z)=\frac{1+z}{1-z}$.
**$z=\frac{w-1}{w+1}$.

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 $\psi(w)$, whose values on the unit circle are given by (1). I am stuck here. I know that I have to use the definition of $\psi$: $\psi=\phi\circ f^{-1}$, 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.

Last edited: Oct 10, 2005
2. Oct 11, 2005

### CarlB

Ooooooo! This is a good one!!!

Here's the direction I'd go in. BUT it doesn't look easy.

$$f(1,\theta) = \sin(\theta)/2$$

$$\frac{df}{d\theta} = \cos(\theta)/2$$

$$\left( \frac{d}{d\theta} + i\right) f(z) = z/2$$

From there, you have to fix the LHS to make it analytic. Of course you only have specified one of the derivatives with respect to r and theta and this is the freedom you need.

Good luck. Do report how it goes.

Carl

Last edited: Oct 11, 2005
3. Oct 14, 2005

### sigmund

CarlB,
Thank you for answering. I did not figure out the problem myself, but my teacher has provided a solution for this problem. See page 3 of the following, where the problem has been solved: http://www2.mat.dtu.dk/education/01141/S/7homework05.pdf.
Actually, I did not really figure out, what you did. Could you therefore, in detail, explain it once more? It would be nice if you could comment on my teachers solution too.

4. Oct 14, 2005

### CarlB

Looking back, I see that I should have used something other than "f" as you were already using it and this is more than confusing. I should have used \psi.
Let me try again...

given $$\psi(r,\theta)$$ we have that

$$\psi(1,\theta) = \sin(\theta)/2$$ and so

$$\frac{d\psi(1,\theta)}{d\theta} = \cos(\theta)/2$$

You can combine these together to give an analytic thing on the right hand side:

$$(\frac{d}{d\theta} + i)\psi = \cos(\theta)/2 + i\sin(\theta)/2 = z/2.$$

Now $$\psi$$ is an analytic function. You know the derivative of psi in the theta direction but you don't know the derivative of psi in the r direction. That's okay cause analytic functions have a relation between their derivatives in r and theta (or between their derivatives in x and y when writing z=x+iy). So you should be able to convert the problem into an analytic differential equation and solve that. But it gets sticky.

The instructor used a nice trick. I suggest doing it his way.

Carl

Last edited: Oct 14, 2005