How can I find all analytic functions f=u+iv with u(x,y)=(x^2)+(y^2)
Thanks for the help. I appreciate it.
ok, so you know that du/dx = dv/dy and du/dy = -dv/dx
you have u so compute it's partials
du/dx = 2x
du/dy = 2y
so dv/dy = 2x and thus v = 2xy + f(y)
dv/dx = -du/dy = -2y so v = -2xy + g(x)
2xy + f(y) = -2xy + g(x)
4xy = g(x) - f(y), and you can see that this has no solution
I should be a bit more careful. You need more than satisfying the C-R equations to declare a function is analytic at a point. The first partials must exist and be continuous at that point. The first partials of (x^2+y^2) + i*0 do exist and are continuous everywhere and thus are at (0,0).