# 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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### Holomorphic function between $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ and $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$

Does there exist a holomorphic function $h$ that sends the set $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ to the set $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$? I tried proving it but I could not. Thanks ...
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### Analytic continuation of $\sum_{n=0}^{\infty} (E_n)^{-s}$

Suppose $E_n$ is a monotonically increasing sequence. Under what conditions on $E_n$ may the sum $$q(s)=\sum_{n=0}^{\infty} (E_n)^{-s}$$ Be analytically continued from $q(s)$ to $q(-s)$. How would ...
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### A problem about proving some spaces are not conformally equivalent

Consider the unit disk $D$,the complex plane $C$ and extended complex plane $C^{\ast}$.Show that no two of them are conformally equivalence. From Liouville theorem,it's easy to see that disk is not ...
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### A property about a bounded harmonic function

Suppose $f(x,y)$ is a bounded harmonic function function in the unit disk and $f(0,0)=1$. Show that $$\iint_{D}f\left(x,y\right)\left(1-x^{2}-y^{2}\right)dxdy=\dfrac{\pi}{2}$$ I don't understand why ...
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### Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
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### Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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### Inversion of lines and circles using explicit parametrizations

Is there a way to parametrize a line and a circle in the complex plane [by $z = z(t)$], to show that under the inversion function $f(z) = 1/z$, a line is mapped either to a line or a circle, and a ...
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### Double periodic entire function

Suppose f is entire and $f(z)=f(z+1)=f(z+\pi)$. Does this imply $f$ is constant? I want to prove that it is constant.I see that it is enough to consider the value of $f(z)$ in between the lines $z=1$ ...
Suppose $G\subset\mathbb{C}$ is open and connected,let $\left\{ f_{n}:n=1,2\ldots \right\}$ be a uniformly bounded sequence of holomorphic functions on $G$ that convergences uniformly on compact ...
A continuous map $f:X\to Y$ is called proper map if for every compact $K\subset\subset Y$ the set $f^{-1}(K)$ is compact. Now, if $\mathbb D=\{z\in \mathbb C;|z|<1\}$. Why the map \$f:\mathbb D\to ...