# 2015-16 Complex integral

Evaluate the following integral for $$z \in \mathbb{C}^+$$.

$\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.$

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# 2014-19 Two complex numbers

Prove that for two non-zero complex numbers $$x$$ and $$y$$, if $$|x| ,| y|\le 1$$, then $|x-y|\le |\log x-\log y|.$

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# 2012-12 Big partial sum

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that $\bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.$

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# 2011-15 Two matrices

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.

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# 2010-9 No zeros far away

Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of $$f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}$$ are in the disk |z|<M on the complex plane.

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