# Tagged Questions

The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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### $\int_C \frac{\log z}{z-z_0} dz$ - Cauchy theorem with $z_0$ outside the interior of $\gamma$

Let the domain $O=\mathbb{C}-(-\infty,0)$, the point $z_0 \in O$ and the circle $\gamma=C(0,r<|z_0|)$ in the positive direction. Compute $\int_C \frac{\log z}{z-z_0} dz$. How do I solve the ...
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### Solve $\cos z=2$ in $\mathbb{C}$

Solve $$\cos z=2 \qquad z \in \mathbb{C}$$ I consider $z=x+i y$, so: $$\cos(x+iy)=\cos(x) \ \cos(iy)-\sin(iy) \ \sin(x)=\cos(x) \ \cosh(y)-i \ \sinh(y) \ \sin(x)$$ I have to satisfy this ...
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### proving a function has a primitive

If a; b are distinct complex numbers let $z(t) = (1-t)a + tb$; $0 <= t<= 1$, be the segment joining a to b. (a) Prove that, outside this segment in the complex plane, the ratio z-a/z-b is ...
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### How can I justify the swap of the limit with the contour integral?

I want to show that if $f$ is analytic then $$f^{''}(w) = \frac{1}{\pi i}\oint_C \frac{f(z)}{(z-w)^3}\,dz.$$ Consider the following: \begin{align} f''(w)&=\lim_{h\rightarrow 0} \frac{f'(w+h)-f'(...
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### How to prove this complex inequality elegantly?

Question Let $z_{1,2}\in U(0,1)\subset \Bbb C$, prove that $$\frac{|z_1|-|z_2|}{1-|z_1||z_2|}\le\left|\frac{z_1+z_2}{1+\overline{z_1}z_2}\right|\le\frac{|z_1|+|z_2|}{1+|z_1||z_2|}$$ Actually I haven'...
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### Fundamental theorem of calculus for analytic functions.

I have to solve the following problem but I have no idea where to start. Any hint or suggestions could be really helpful, thanks! Suppose the continuous functions $f(e^{i \theta})$ on the unit circle ...
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### How to prove $|y|\le |\sin(z)|\le e^{|y|}$ in complex analysis where $z=x+iy$?

How to prove $$|y|\le |\sin(z)|\le e^{|y|}$$ in complex analysis where $z=x+iy$. I don't think I need the entire solution. May be just a set up or approach. I started with sin(z) formula and reached ...
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### Find the laurent series for $\frac{1}{z(z-2)^2}$ centered at z=2 and specify the region in which it converges.

My attempt: $$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$ This is ...
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### Inverse of the Cross Ratio for Mobius Transformation from Circle to Circle

I'm reading Conway's complex functions of one variable, and in chapter 3 he goes over Cross-Ratios. He defines the cross ratio to be $(z,z_1,z_2,z_3)=\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, where ...
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### Criticize my math at finding radius of convergence of $\sum_{n=0}^{\infty} a^{n^2}z^{n}$
$\sum_{n=0}^{\infty} a^{n^2}z^{n}$ Let R be the radius of convergence. Then we have \$\frac{1}{R} = \lim \sup |a^{n^2}|^{1/n} = \lim \sup |a|^{n} = \lim_{n \rightarrow \infty} [ \sup \{|a^{n}|,|a^{n+1}|...