Is there a function that is analytic in the complex plane.
If so determine all the functions f.
So i know that if u(x,y) and v(x,y) are harmonic and satisfy the cauchy riemann equations f are analytic.
My standard method for this kind of problems is:
Find then with CR(1): i integrate so i get v(x,y) = something + g(x).
Then with CR(2): i get g(x).
But with my method it gets really messy! i cant solve the integration, so i wonder if theres another way to do this?
Maple gives the following result. So according to this, V(x,y) is the following function plus some g(x).
Now.. constructing f(z) from that is a problem of a whole new level, lol. But the complexity of this doesn't preclude the possibility of there being a neat trick. Maybe f(z) collapses into something simple.
This is actually not that complicated. I guessed that there must be some "symmetry" between the hyperbolic/trigonometric sines and cosines, as usually exhibited by the real and imaginary parts of an analytic function, and indeed, if U(x,y) is defined as in the initial post, then
The above post is maple expanding that into complex exponentials for some reason. But the C-R equations do equal for these functions, try them.
To find f(z) from these, remember that if two analytic functions coincide on any line, say on the real axis, they coincide everywhere. So if you express u + iv as a function of a real variable, by treating y=0, and x = z, you can get the general equation for f(z). Note that on the real axis, the imaginary part will vanish, because sinh(0) is 0. Also, cosh(0) = 1. So from this you can get:
Where c is any constant. We need that there because given the real part of an analytic function, the imaginary part can vary up to a constant added/subtracted. Technically I should have put it there when writing out V(x,y).