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

\[

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

\]

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

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

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}=A^{n}+B^{n}.

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.