Write a complex valued function in terms of z

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The discussion focuses on expressing complex valued functions in terms of the complex variable z. Specifically, it examines the function f(z) = 1/z, derived from f(x,y) = (x/(x^2+y^2)) - i*(y/(x^2+y^2)). Key insights include the importance of recognizing relationships such as x^2 + y^2 = |z|^2 and x - iy = z*. The conversation emphasizes that not all functions can be neatly represented in this manner, highlighting the complexity of certain transformations.

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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?
 
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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.
 

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