Write a complex valued function in terms of z

[AlonsoMcLaren]

Is there a general way of writing a complex valued function
f(z)=u(x,y)+iv(x,y) in terms of z?

For example, suppose that I know

f(x,y)=(x/(x^2+y^2))-i*y/(x^2+y^2)

How do I know that it is actually f(z)=1/z?

 

[mfb]

You can express x and y as function of z, but that will give really ugly formulas.

It is useful to recognize some formulas (like x^2+y^2=|z|² = zz* and x-iy = z*), and if those don't work you can still guess how the function could look like.
Not every function has a nice representation like your example.