# 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, ...

102 views

### Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$\sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha}$$ ...
164 views

### Schwarz Reflection Principle vs. Analytic Continuation

Analytic continuations are unique on simply connected domains: $$F,F':\Omega\to\mathbb{C}:\quad F\restriction=F'\restriction\implies F=F'$$ Schwarz reflection principle offers analytic continuations ...
44 views

182 views

### Prove there is no branch of arg $z$ on $0 < z < 1$.

If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$. ...
45 views

27 views

80 views

### A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
83 views

### Find the real and imaginary part of the following

I'm having trouble finding the real and imaginary part of $z/(z+1)$ given that z=x+iy. I tried substituting that in but its seems to get really complicated and I'm not so sure how to reduce it down. ...
53 views

### Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
681 views

### Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
78 views

Find annulus of convergence of Laurent series $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^3}$ My answer: $0<|z-i|<\infty$ $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^2}$ My answer: $|z-i|&... 2answers 145 views ### how to write eqn of line in complex form Write the given equation of a straight line in complex notation: Straight line through 1 and (-1 - i) Attempt: So i treated this initially just like a set of coordinates in the set of R thus (1,0) ... 1answer 36 views ### Prove the inequality$| \frac{z}{|z|}-1| \leq |arg(z)|$Prove the inequality$| \frac{z}{|z|}-1| \leq |arg(z)|$Here is what I got$z=r(cos \theta +i sin \theta)$. So$LHS= ((\cos \theta -1)^2 +sin^2 \theta)^{1/2}=(2-2\cos \theta)^{1/2}$Note that$-1 \...
Evaluate: $$\lim_{z \to 0} \psi(-z)\cdot \bigg ( 1 - 2z(z+1) \bigg) - z\cdot\psi'(-z)$$ If we simply substitute in $0$ that gets us infinity, and problems. The answer is $-2 - \gamma$ How do we ...