# Questions tagged [branch-cuts]

A branch cut is curve in the complex extending from a branch point of the function.

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### $f(z)$ is holomorphic everywhere in the complex plane. It is positive for $z \in \mathbb{R}$. Can I choose $\log(f(z))$'s branch cuts to avoid $z=0$?

Suppose that $f(z)$ is holomorphic everywhere, and it's real and positive on the real axis (specifically, $f(z) > \epsilon > 0$ for all $z \in \mathbb{R}$). I want to define an appropriate ...
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### Radius of convergence at $2+i$ of $f_i(z)=\frac{1}{\phi_i-2^{1/4}}$ with $2^{1/4}$ being the positive real root

Let $\phi_k(z), k=0,1,2,3$ the branch cuts of $z^{1/4}$. Consider $$f_k(z)=\frac{1}{\phi_k-2^{1/4}}, \quad 2^{1/4}=|2|^{1/4}e^{i0}>0$$ Find the radius of convergence of the series expansion at $2+i$...
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### Series expansion of $z^{1/3}$ at z=1

Obtain the series expansion of $f(z)=z^{1/3}$ at z=1 such that $1^{1/3}=\frac{-1+i\sqrt{3}}{2}$ The way I've done it is the following: I need $1^{1/3}=e^\frac{i\arg{1}}{3}=e^{i2\pi/3}$, so any branch ...
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### Branch cut in integral function

I'm not very versed in complex analysis and I'm trying to understand some concepts on branch cuts and contour integration. Consider a function $$I(s)=\int_0^1 d\alpha\ \frac{1}{f(s,\alpha)},$$ such ...
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### Riemann surface of the hypergeometric function

The hypergeometric function $_2F_1(a,b,c,z)$ has a branch cut extending from $z=1$ to $z=\infty$. Does this define an infinite-sheeted Riemann surface (like that for $\log{z}$) or one with a finite ...
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### Analytically continuing a function defined via an integral

Suppose we want to analytically extend a function $g(z)$ that is defined as an analytic function everywhere in the complex $z$ plane except the negative $z$ axis (for example, via an integral). Would ...
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### What is meant by saying that $[f(z)]^p$ has a holomorphic branch in a neighborhood of $z_0.$

Let $\Omega \subseteq \mathbb C$ be a domain in the complex plane and $f \in \text {Hol} (\Omega).$ Let $z_0 \in \Omega$ be such that $f(z_0) \neq 0.$ Then for any $p \gt 0$ there exists a ...
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### What does the argument limitation $| \arg(z)| < \frac{3\pi}{2}$ mean for the asymptotic development of $E_1(z)$?
Given an asymptotic of the $E_1(z)$ complex function (from Gradshteyn & Ryzhik p. xxxv): $$E_1(z) \sim \frac{e^{-z}}{z}\left[1 -\frac{1}{z} +\frac{2}{z^2} - \ldots \right]$$ Where is this ...