# 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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### Comparison of entire functions with $f=cg$

Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$. If $g\equiv 0$ then we ...
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### Convergence of a series of complex numbers.

Let $f : \mathbb C \to \mathbb C$ be a non constant entire function. Does the series $\sum_{n=1}^ \infty \frac{1}{n} f(\frac {z}{n})$ converges at any point $z \in \mathbb C$ ? I think this will not ...
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### How to derive this Hankel's Contour integral formula with gamma function?

This relation was put up in The Art Of Computer Programming and no derivation was offered. Please help me understand this better. $$\frac{1}{\Gamma (z)} = \frac{1}{2i\pi} \oint\frac{e^t dt}{t^z}$$ ...
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### topology of compact convergence, closed sets

Let $H(\mathbb{D})$ be the vector space of all analytic functions on the unit disk. Then the topology induced by uniform convergence on compact subsets is metrizable. Thus the following topology ...
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### About some good references for self study

I'm willing to start a self study of Hardy spaces, Bergman spaces and Bloch spaces. I would like to know good books on the subject. Since I'm going to study on my own, would be great to find one that ...
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### Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
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### Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
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### Finding the Laurent series of a function

I'm trying to work through the following example: Find the Laurent series of: $$f(z) = \frac{1}{z(z-2)^3},$$ about the singularities $z = 0$ and $z = 2$ (separately). Hence ...
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### Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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### Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
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### Contour Integral involving hyperbolic functions

I would like to evaluate: $\displaystyle\oint_C \frac{e^{4z}-1}{\cosh(z)-2\sinh(z)}\,\mathrm dz$ where $C$ is a unit circle in the complex plane and $z=x+iy$. I did not find any singular ...
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### Does this function Contradict this colloary?

Colloary: If $G$ is a domain and $f: G \to \mathbb{C}$ is analytic and not identically zero, then the zeros of $f$ are isolated. If the domain $G$ is closed and bounded, then the zeros are finite in ...
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### Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
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### How to determine Laurent series associated to $f(z)$ [closed]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
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### Prove that these are the automorphisms of the Poincare upper half plane

If $P = \{Im(z) > 0\} \subset \mathbb{C}$, prove that $f:P \rightarrow P$ is an automorphism iff $f(z) = \frac{az+b}{cz+d}$, for $a,b,c,d \in \mathbb{R}$, with $ad-bc > 0$. I had thought about ...
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### Does $f(z)$ has an essential singularity at $z=z_0$?

Let $f(z)=g(z)h(z)$ such that $g$ has an essential singularity in $z=z_0$ and $h$ is holomorphic in a neighbourhood of $z_0$ then $f$ has an essential singularity in $z=z_0$? Im trying to see this ...
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### Elementary fact about holomorphic function

I found some assertion in this article (bottom of third page) of François Trèves which in a way states that taking a bounded simply connected domain $D$, $H$ the open upper half plane, a holomorphic ...
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### contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
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### How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
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### Existence of analytic function from A to B . [duplicate]

Does there exists a non constant analytic map $f:A\to B$ . Where $A=\{z\in \mathbb C~:~ |z|\neq 0\}$ and $B=\{z \in \mathbb C ~:~ |z|>1\}$. I am unable to construct one
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### Minimum modulus principle - looks like a counterexample?

The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
Let $A=\{z\in C ~:~ |z|>1\}$ , $B=\{z\in C ~:~z\neq 0\}$. Which of the following is true? 1.There exists a continuous onto map $f:A\to B.$ 2.There exists a continuous one to one map \$ f:B\to ...