1) where

By definition,

By definition,

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- Jun 19th 2007, 01:33 PM #1degomanGuest

- Jun 19th 2007, 02:15 PM #2

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- Jun 19th 2007, 02:23 PM #3

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- Jun 19th 2007, 02:33 PM #4

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4)I think you made a mistake. That is not the solution to:

5)The boring what is to substitute that into the PDE and verify that it solves it, i.e. adds up to zero. A more exciting way is to use the Cauchy-Riemann equations. The prinipal logarithm is holomorphic on . That means are harmonic functions.

We can write,

That means the real part is solves Laplace's equation.

- Jun 19th 2007, 03:00 PM #5

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3)I am really not sure about this one, but I think I know what I am doing. First, in your problem you have the Dirichlet problem in 3 variables, is that a mistake? I will do it in two variables.

Seperate variables and write .

Then by substituting this into the PDE we have:

Divide through by to get:

And hence by the seperation of variables techinque we have a system of ODE:

Look at the first equation together with the boundary value problem implies that for means non-trivial solutions can only exist iff , that is the solutions are sines and cosines.

Look at the second equation by similar reasoning . So if (by the first equation) it**must**mean that for otherwise and that will lead to trivial solutions to the two point boundary value problem.