MHB Billy 's question at Yahoo Answers (Complex analysis)

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The function f(z) = z + 1/z can be expressed in terms of polar coordinates as f(z) = u(r, θ) + iv(r, θ). By substituting z with re^(iθ), the function simplifies to f(z) = re^(iθ) + (1/re^(iθ)). This leads to the real part u(r, θ) being (r + 1/r)cos(θ) and the imaginary part v(r, θ) being (r - 1/r)sin(θ). The discussion provides a detailed breakdown of the transformation from Cartesian to polar coordinates for complex analysis. This method effectively illustrates the relationship between the function's components in the complex plane.
Fernando Revilla
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Here is the question:

write the function f(z)=z+1/z (z cannot = 0)
in the form f(z)=u(r,theta) + iv(r,theta)

Here is a link to the question:

Complex analysis help? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Billy,

Using $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ and if $z\neq 0$ (i.e. $r\neq 0)$:
$$\begin{aligned}f(z)&=z+\frac{1}{z}\\&=re^{ i\theta}+\frac{1}{re^{i\theta}}\\&=re^{i\theta}+ \frac{1}{r}e^{-i\theta}\\&=r(\cos \theta+i\sin\theta)+ \frac{1}{r}(\cos \theta -i\sin\theta)\\&= \left (r+\frac{1}{r}\right)\cos\theta +i\left (r-\frac{1}{r}\right)\sin\theta\end{aligned}$$
 
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