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 of a complicated function

I would like to study the branch points of the function $q(z)=\frac{z^{\frac{1}{3}}+\frac{1}{2}z^{\frac{-1}{3}}}{[1+z^{-\frac{2}{3}}]^{\frac{1}{2}}}$ . I plotted its real part with Mathematica and it ...
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Proving that : $Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$ is single valued.

The Legendre function of second kind $Q_\nu(z)$ has branch points at $z=\pm 1$. The branch points are joined by a cut along the real axis. Show that $$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$...
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How to prove there is a branch point in complex function?

I'm working with the function $$f(z)=(-z)^{1/3}$$ And I've been asked to prove that the function has a branch point at x=0 by calculating the difference: $$\frac{f(x+i \theta)-f(x-i\theta)}{r^{1/3}}$$ ...
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Are there a, in depth, classification of branch points in complex analysis?

Disclaimer: I have posted this question on mathoverflow.net following the instructions of this topic. In complex analysis we have well know results about isolated singularities. Poles are ...
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How to determine if $z+\sqrt{z-1}$ and $\frac{\sin{\sqrt{z}}}{\sqrt{z}}$ is a multi-valued function?

What I know: I understand that for a complex function to be a multi-valued funciton they must have some branch points. I understand branch points as this definition (translated into English from my ...
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REVISED Proving $-1,1$ are branch points of $\sqrt{z^2-1}$

On some lecture notes that I am working on there is an exercise to prove that $-1,1$ are branch points of the multi-function $\sqrt{z^2-1}$. I know that the branch $f=\sqrt{rs}e^{i\frac{1}{2}(\theta_1+...
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51 views

Showing that contour integral around branch point converges to zero

I have encountered several times where I need to calculate a contour integral around a circle or half circle with an infinitesimal radius that is enclosing a branch point. In the cases I have ...
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53 views

Branch cuts imaginary part

This is a very simple gap in my intuition about branch cuts. I have heard informally a statement of the type: if a complex function suddenly acquires an imaginary part at the point $x_0$ in the real ...
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60 views

Inverse Laplace transform through contour integration

$$ F(s)=\frac{\sin(s)}{\sqrt{s}} $$ Does the inverse laplace transform of this function exist? But how do we find this through contour integration? Any suggestions please. Because as this is special ...
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Prove that a point it's a branch point

I'm having doubts about how to approach this problem. I have a function $f(z)= \sqrt{z^2-1}$, where the argument of the complex inside the square root it's $\arg{(z^2-1)}=\arg{(z-1)}+\arg{(z+1)}$ and ...
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Branch cuts for $f(z)=\sqrt[3]{z^3+1}$

I'm struggling as to how I'm supposed to do an analysis of how different branch cuts would affect a function f(z). The problem I'm struggling with has $f(z)=\sqrt[3]{z^3+1}$ I've found the branch ...
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Riemann Branch Points and Covering

I am currently struggling with understanding how to find the branch points and behavior of covering of implicitly defined irreducible functions $P(x,y)$. An arbitrary example I am working on is to ...
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Find residue of $f(z) = e^{\frac{1}{1+√z}}dz$ in the point $z = 1$ for all branches of $e^{\frac{1}{1+√z}}$.

As is said in the title I would like to evaluate the residue of $f(z) = e^{\frac{1}{1+√z}}dz$ in the point $z = 1$ for all branches of $e^{\frac{1}{1+√z}}$. This is new for me, but after reading some ...
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Find a function that has multiple branch cuts and state why those branch cuts remove the discontinuities.

I am trying to find a function where multiple branch cuts are required. One example I've encountered online was $\sqrt{(z)}\sqrt{(z-1)}$; however, I am confused as to why the branch cut $y = 0, x \leq ...
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Branching points of a multi-valued function

Consider $g(z) = w $, where $w^3 - 3w -z = 0 $( $g$ is the inverse of $f(z) = z^3 - 3z$). What are the branching points of $g(z)$? Sketch the scheme of the Riemann surface of $g(z)$ I know that the ...
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Finding analytic continuatiuon of a branch

I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases. The ...
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Working with branches of complex functions

Let $f(z)$ be the principal branch of $z^{1/2}$ on $\mathbb{C}\setminus (-\infty,0]$, i.e. we insist $\theta=\arg(z)\in(-\pi,\pi)$ and define $f(z)=\sqrt{|z|}e^{i\theta/2}$. Let $g(z)$ be an analytic ...
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Choosing Branch cut for the function $f(z)=\sqrt(z-a)\sqrt(z-b)$ for general $a,b\in \Bbb C$

In the example of Wikipedia: https://en.wikipedia.org/wiki/Branch_point#Branch_cuts they use the function $f(z)=\sqrt z\sqrt {z+1}$ and said that the branch cut of this function is the section $[0,1]$ ...
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Find $\oint_{|z|=1}\log\left|\frac{1}{1+z^p}\right|\frac{dz}{z} $ where $0<p<\frac{1}{2}$

Find $$\oint_{|z|=1}\log\left|\frac{1}{1+z^p}\right|\frac{dz}{z} $$ where $0<p<\frac{1}{2}$ Attempt $$I=\oint_{|z|=1}\log\left|\frac{1}{1+z^p}\right|\frac{dz}{z} $$ Define, $$f(z)= \frac{1}{1+z^...
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Let $\log$ be a branch of the logarithm in $D$ such that $\log e=1$. Find $\log e^{15}.$

Let $D\subset\Bbb C$ be the complement of the closed spiral $\{e^{\theta+i\theta}:\, \theta\in\Bbb R\}\cup\{ 0\}$. Let $\log$ be a branch of the logarithm in $D$ such that $\log e=1$. Find $\log e^{15}...
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Square root of complex logarithm

Question: Let the square root be computed using the branch $L_{\pi/2}$ of the logarithm. Where is $(z^2-i)^\frac{1}{2}$ not analytic? My solution: The function is not analytic when $\sqrt{z^2-i}=yi, y ...
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Are contours allowed to contain branch points / branch cuts on the contour (not inside)

I just wanted to verify something, since I saw some notes online that confused me. They were doing some contour integral and $0$ was the only branch point of the function, and they chose the branch ...
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How to Prove That Branch Points Of $\log (f(z))$ are Zeros and Poles of $f(z)$?

In many books and notes, It is written that the branch points of $\log (f(z))$ are points $z\in \Bbb C$ such that $f(z)=0$ or $z$ is a pole of $f$. Here we assuming that zero of $f$ is the sense of ...
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83 views

Is infinity a branch point of $\sin(\sqrt{z})$?

Does the function $f(z) = \sin(\sqrt{z})$ have a branch point at infinity? I'm confused because infinity is an essential singularity of $\sin(z)$, so I'm not sure how to do the usual $z\to w=1/z$ ...
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37 views

how can I find the branch points and cut?

For $f(z)=\sqrt{z^3+8}$, I need to find all branch points and branch cuts, not in the disk $|z|<2$, which make the function single-valued. Here I started with $z=-2,1-i\sqrt{3},1+i\sqrt{3}$ are ...
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How can we know if a solution of differential equation is multivalued without solving it?

For a differential equation, say $q(x)y''(x)+p(x) y'(x)+s(x)=0$, how could we know if its solutions are multivalued without solving it. Could we study the branch points and branch cut by simply ...
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Complex Analysis - Defining a branch cut for a function with multiple branches which DON'T lie on an axis

For the function: $$f(z)=(z+\sqrt{3})^{1/2}\ln{(z-1)}$$ with the branch of this function chosen such that $$-\frac{4\pi}{3}<\arg{(z-1)}\leq\frac{2\pi}{3}$$ and $$-\frac{\pi}{2}<\arg{(z+\sqrt{3})}...
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Complex Integral $\int_{-\infty}^{+\infty}\frac{e^{(-A\sqrt{x^2+B}+Cix)}}{\sqrt{x^2+B}}$

I am confused calculating the below integral: $$\int_{-\infty}^{+\infty}\frac{e^{(-A\sqrt{x^2+B}+Cix)}}{\sqrt{x^2+B}}dx$$ Where A, B, C are real and A and B are positive. I don't think I can use roots ...
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42 views

Branch point of the given function

Prove or disprove that $0$ is a branch point of $f(z)=\frac{e^{\sqrt{z}}-e^{\sqrt{-z}}}{sin{\sqrt{z}}}$ can I consider $z=\epsilon e^{i\theta}$ (points on a small circle centered at $0$) to determine ...
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55 views

Singularity structure of a multivalued function

Consider the function $$f(z) = \frac{1}{z} \ln \left( \frac{1-z}{1+z} \right).$$ This is clearly multivalued. There has to be two branch points at $\pm1$. Are there any other singularities, such as ...
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154 views

Cauchy's residue theorem extended to branch cuts

I'm familiar with Cauchy's residue theorem for calculating integrals in the complex plane. I'm wondering if there's a natural way of extending this to functions which also contain branch cuts. As an ...
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132 views

Does the dilogarithm function (which is multi-valued) have a single-valued inverse?

The $p$-logarithm is defined for $|z|<1$ by $$\text{Li}_p(z)=\sum_{n=1}^\infty\frac{z^n}{n^p}$$ and defined elsewhere in $\mathbb C$ by analytic continuation, though it may be multi-valued, ...
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Can a function have a branch cut along the real and also imaginary axis?

Is it possible for a complex function to have a branch cut along the real axis and also the imaginary axis, that cross over like a + sign?
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Traveling Around a Complex Number

In functions with complex arguments it's been said that to determine whether a point is a branch point or not we have to travel around that point and see if we get the same answer. How do I travel ...
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Interpretation of branch cut

In wikipedia, I found that a branch cut can be thought of as a sum of infinitely many poles arranged along a line in the complex plane with infinitesimal residues. And wiki takes $ln(\frac{z+1}{z-1})$ ...
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Prove set of branch points is discrete assuming...

Let $X$ and $Y$ be connected Riemann surfaces. Let $F: X \to Y$ be holomorphic non-constant (and thus open and continuous). Denote the set of ramification points $Ram(F) :=\{p \in X \ | \ mult_pF \ge ...
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$arctan(z)$ and Riemann Surfaces

How do I link the complex trigonometric function arctan(z) and it's branch cuts and branch points with Riemann surfaces? I have seen the picture of the function's Riemann surface but I don't ...
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Smoothness of the geometric mean of $W_0(x),\, W_{-1}(x)$ for real $x<0$

$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$ This question is closely related to the similar one about the ...
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Odd number of branch points possible for a multivalued complex function?

Is it possible for a multivalued complex function $f:\mathbb{C}\to\mathbb{C}$ to have an odd number of branch points? I'm asking, as I've only seen examples of multivalued functions with an even ...
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Set of branch points isn't discrete, but branch points are isolated?

I refer to Chapter II.4 of Rick Miranda - Algebraic curves and Riemann surfaces, which I understand says that the branch points of a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $...
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Evaluating the following integral $\int_{0}^{\infty} \frac{\ln(x^{2}+1)} {(x(x^{2}+1))} dx$

I have tried many different methods on this particular integral, none of them yielding any fruitful results. Here was an attempt I(t) = $\int_{0}^{\infty} \frac{ln(tx^{2}+1)} {(x(x^{2}+1))}dx$, I(1) ...
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Weierstrass-type factorization (reconstruction) of a function with branch cuts

Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,...
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Integrating a function with two branch point $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $

I was studying a book about residue intigrating then I get into trouble solving this : $ {\int_{1}^\infty} \frac{dx}{{x(x^2-1)}^{1/2}} $ I know that I have to choose a contour like this: but I don't ...
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Proof Check: Complex function and Branch points

requesting a proof (rationale) check as i'm unsure whether or not my rationale is legitamate or not. Thanks in advance Consider the function $$f(z) = \frac{\omega}{(z^2 + \omega^2)e^{\alpha\sqrt{z/d}}...
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87 views

Branch cuts of complex logarithm for Friedel oscillation

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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60 views

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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489 views

Finding the branch points of $\log(z^2-1)$

This example is presented in the following paper (http://math.mit.edu/classes/18.305/Notes/n00Branch_Points_B_Cuts.pdf, example 2.2, page 11). The author uses the identity $\log(z^2-1)=\log(z+1)+\log(...
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303 views

How to solve this integral with contour integration $\int ^{\infty }_{0}\frac{\ln( x)}{( x+1)^{\alpha}}\,dx$

I know how to solve this integral without contour integration. The answer to the integral is $$\int^{\infty }_{0}\frac{\ln( z)}{( z+1)^{\alpha}} \,dx=\frac{H_{\alpha-2}}{1-\alpha} ,\; \alpha>1,$$ ...
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64 views

Why is it the case that infinity is also a branch point for log

I thought about considering $L(z)=log|\frac{1}{z}|+i\theta$ and consider the behaviour as $z$ tends to $0$? All I am observing now is that the function tends to infinity with it. I am really stuck on ...
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318 views

Branch points of $f(z)= \frac{\sqrt{z} \log(z)}{(1+z)^2}$

How does one go about finding the branch points/holomorphic branches of a multi-function composed of several other multi-functions? Here is an example of what I mean: Let $f(z)= [\frac{\sqrt{z} \...