# Tagged Questions

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### How to find out the real part of this.

I have to sum this: $$S:=\cos\left(\frac{\pi}{M+1}\right)+\dots+\cos\left(\frac{M\pi}{M+1}\right)$$ Where $M$ is a given natural number. I tried with this: Since $$e^{i a}=\cos a+i\sin a$$ and ...
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### Find the residues of $f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}}$

Consider the function $$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$ I want to calculate the residues of $f$ in $\{+i,-i\}$. Using the usual techniques, we have that ...
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### Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
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### Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
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