Domain and range of complex functions

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Main Question or Discussion Point

If I am given a function:
f(z) = w(x,y) = u(x,y) + iv(x,y)

How do I find the domain and range of that function? Are there any good lectures online on this?

I know I have to use two planes; One for u and v, and one for x and y. Besides that, I am lost.
 

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haruspex
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The domain is easy - just the intersection of the domains of u and v. The function f is defined at (x, y) if and only if both u and v are.
The range is tougher. Knowing the ranges of u and iv separately gives you an 'upper bound' on the range (namely, the direct product of the two separate ranges) but there will likely be many combinations of values that can't arise.
If u and v are differentiable then there will be an interesting equation for the boundary of the range. Can't come up with it immediately, and couldn't find anything online... will try to derive it later.
 
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haruspex
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OK, I think I have the range boundary formula. It's where the Jacobian determinant is zero:
ux vy = uy vx
E.g. consider f = (1+x2) (cos(y) + i sin(y)). The formula collapses to x = 0, i.e. f = cos(y) + i sin(y), the unit circle.
 
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The domain and the range of a complex function is a 2D region each. For example, a circle. For many important functions, the domain and the range is the entire complex plane, with a finite or countable number of singular points.
 

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