This is nothing to do with Laplace transforms.
Originally Posted by jezzyjez
Laplace's equation is this:
Cheers guys got the answer to the first part now.
Not to difficult once you get your head round what all the words mean.
You can find the harmonic conjugate by the following expression:
Yeah, I know it looks intimidating. Start from the inside and work out and when you do an integration, drop the constant of integration. I'll do the first part for then:
see, I'm starting from the most inner level. Now, I want to partially integrate that with respect to y:
Now, I want to take the partial of that with respect to x. See, that's the first three steps inside the parantheses. Now get the partial of u with respect to y, negate them, then integrate the bunch with respect to x. See, got that whole parentheses thing now. Do that last integral, bingo-bango.
Edit: Oh yeah, I forgot to mention don't use that expression unless you can go through the steps in deriving it. Any Complex Analysis text book goes over finding the complex conjugate. I just combined all the steps into one expression for fun.