As far as I am aware it is known that for any complex analytic function, the gradient of the real part of the function and the gradient of the imaginary part of the function are at right angles.
For example
$f(x+iy)=(x+iy)^2+3(x+iy)+2$,
has $(2x+3,-2y,0)$ for the gradient of the real part and $(2y,2x+3,0)$ for the gradient of the imaginary part, and these vectors are clearly orthogonal.
My question is, is there a class of functions which exists, where the angle between the gradients of the real and imaginary parts of the function is arbitrary?
Or more specifically, if I choose an angle, is it possible to find a general class of functions which will produce gradients for the real and imaginary parts which are seperated by this angle?
I hope this isn't too badly explained.
Any help or suggestions would be appreciated. Thanks. :)
