# harmonic function

by Tony11235
Tags: function, harmonic
 P: 276 I am to find a function U, harmonic on the disk $$x^2 + y^2 < 6$$ and satisfying $$u(x, y) = y + y^2$$ on the disk's boundary. I am not sure where to start. Hints, help, anything?
 Sci Advisor HW Helper P: 9,398 Use the integral formula.
 PF Gold P: 864 I would think Cauchy's integral formual would be useful here: you have the value of a function on a boudry and want the value in the interior.
P: 76

## harmonic function

You are trying to solve the Laplace equation on a disk. Try seperation of variables, then break it down to 2 ODE's. Here is a start for you..

You will probably need to solve the PDE in polar coordinates.

- harsh
P: 276
 Quote by harsh You are trying to solve the Laplace equation on a disk. Try seperation of variables, then break it down to 2 ODE's. Here is a start for you.. You will probably need to solve the PDE in polar coordinates. - harsh
Then is $$u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)$$ a boundary condition?
P: 76
 Quote by Tony11235 Then is $$u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)$$ a boundary condition?
Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta

- harsh
P: 276
 Quote by harsh Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta - harsh
I know. In an earlier problem I had to compute the laplacian in polar. Oh and one more thing, is there anything else I need to know about $$\theta$$? Other than $$0 < \theta < 2\pi$$ ?
 P: 76 The theta condition that you are going to use, I believe, will be that theta is 2pi periodic. - harsh

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