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Complex conjugates of functionsby brmath
Tags: complexanalysis 
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#1
Aug2513, 12:36 PM

P: 329

I am looking at a complex function f(z) and want to know something about f(z)f*(1/z*). We could assume for now that f(z) is analytic or at least meromorphic. Are there any identities involving this product? Is there any way to decompose the f*(1/z*) into a function of f?



#2
Aug2513, 03:23 PM

P: 615

Suppose ##f(x+iy)= u(x,y)+iv(x,y)##, where ##u## and ##v## are realvalued functions. Then, we have that ##\bar{f}(\frac{1}{xiy})=\bar{f}(\frac{x}{x^2+y^2}+i\frac{y}{x^2+y^2})=u(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2})iv(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})##. Thus, we have ##f(z)\bar{f}(\bar{z}^{1})=u(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})u(x,y)+v(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2})v(x,y)+i (u(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}) v(x,y)v(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}) u(x,y))##. 


#3
Aug2513, 06:58 PM

P: 329

Hi, I'll try not to write f(z) when I mean f. I already know that 1/z* = z/z. I had hoped there are some standard indentities or inequalities involving f(z)f*(1/z*).



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