# Questions tagged [branch-points]

A branch point is a point in the complex that can map from a single point to multiple points in the range.

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### Branch Points and Cuts

I am having a lot of problems understanding what exactly branch points are and how they are computed for a function. There is this one problem that I just can't seem to get around to understanding, ...
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### Why is this not a logarithmic branch point?

Good afternoon! I wanted to ask a quick question, as a beginner in complex analysis, I am trying to get my head around branch points. I came upon some lecture notes, but then did not understand ...
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### Branch cuts of $f(z)=\frac{1}{\sqrt{1+z^4}}$

Let $f(z)=\frac{1}{\sqrt{1+z^4}}$. The branch points are $e^{\frac{2k-1}{4}\pi i}$. I am going to find ALL possible branch cuts. When $z$ traces a closed curve (anticlockwise) around any of the above ...
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### Prove $[z^{1/2}]$ has no holomorphic branch on any open ball $B(0,R)$

Prove the multifunction $[z^{1/2}]$ has no holomorphic branch on $B(0,r)$. Here is my attempt: Let $f: B(0,r) /(-\infty,0] \to \mathbb{C}$ be a holomorphic branch of $[z^{1/2}]$. Note $-f$ defines ...
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### Help on finding the branch of a multi-valued function

I'm trying to answer the following question: Find a branch of the following multiple-valued function that is analytic in the given domain: $(z^2-1)^{1/2}$ in the unit disk |z|<1 I tried to answer ...
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### Integrand with branch point

I am trying to learn how to work with branch points. I decided to change the sign in the next classical example: $$f(x)=\int_{-1}^{+1} \frac{1}{\sqrt{1-x^2}}$$ In this example, simply select the ...
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### Residue at $\infty$ for $1/\sqrt{z^2-1}$?

This feels rather silly to ask, but this has been confusing me as of late. One exam question I was attempting recently was to find the contour integral of $1/\sqrt{z^2-1}$ over the contour $\Gamma$ ...
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### Understanding the function $\frac{1}{\sqrt{z^2+11}}$

I want to understand the function: $f(z) = \frac{1}{\sqrt{z^2+11}} dz$. This functions seems to have two branch points $\sqrt{11}i$ and $-\sqrt{11}i$. Does it also have a brunch at $\infty$? Usually ...
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### How to properly understand branches of complex functions

$\DeclareMathOperator{\Log}{Log}$ I have several problems to understand the concept of branches and how to find analytic branches. From what I learned, for example for the complex logarithm, it is a ...
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### Meromorphic function at a point.

i'm study meromorphic function at the complex plane extended, but i have a trouble. I know if $f \colon D \subseteq \mathbb{C} \to \mathbb{C}$ $f$ is meromorphic on D if the singularities of $f$ ...
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### Questions about branch point of holomorphic map

In order to calculate genus of compact Riemann surface using Riemann-Hurwitz theorem, we have to determine the branch points first. Question: For holomorphic maps between $\Bbb{CP^1}$, is there a ...
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### $\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$ difficult integral with two branch cuts

$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$ I defined two branch cuts along the real axis: $[-\infty ,-\frac{1}{a}]$ & $[0,\infty]$ with the following contour: I defined the $arg{(z)} =0$ ...
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### Branch point of $\log(z)+arcsin(z)$

i try to do it by substituting $z$ into $\frac{1}{t}$. Then, it would become this function $-iln[i+t\sqrt{(1-\frac{1}{t^2})}]$. Teacher says that when $z=z_\infty$ is a branch point, only could be ...