# 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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### Complex Analysis Liouville's Theorem [duplicate]

I am not sure how to solve the following problem: Use Liouville's theorem to prove that if f(z) is holomorhpic in the in entire complex plane and $f(z+1) = f(z)$, and $f(z+i)=f(z)$ for all $z$ in $C$ ...
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### Entire function , guidance or advice

Let $f:\mathbb{C}\to\mathbb{C}$ entire and , $|f(z)|\le m\ e^{a\mathop{\rm Re} z}, z\in\mathbb{C},$ $a,m>0$ Show that $f(z)=Ae^{az}, A\in \mathbb{C}$ I think that most of these case are dealt ...
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### Show that the set of one-to-one holomorphic maps $\Bbb{C}\setminus\{a,b,c\} \to \Bbb{C}\setminus\{a,b,c\}$ forms a finite group.

Let $\Omega = \mathbb{C}\setminus\{a, b, c\}$ be the complement of three distinct points in the complex plane. Show that the set of one-to-one holomorphic maps $f : \Omega \to \Omega$ forms a finite ...
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### Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
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### Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

\begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
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### Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$
Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map \$f(z) = ...