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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Sufficient and necessary condition for $(z^3)^\frac{1}{3}=z$

My goal is to give a sufficient and necessary condition for $(z^3)^\frac{1}{3}=z$ for the principal branch of the cube root. Let $z=re^{i\theta}; \, -\pi \leq \theta < \pi; \, z^3=r^3e^{i(3\...
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Contourn integration with gaussian-like argument

I'm currently working on evaluating the following integral: $$ I=\int_{-\infty}^{+\infty}\frac{a-x^2+b}{(a-x^2)^2}e^{-i c x^2}dx, $$ where $a,b,c$ are real and positive constants. I wish to compute ...
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Some exercises about branches

I'm having difficulty with the following questions. I'm also having trouble understanding how to work with branches and branch cuts - it's not very intuitive to me what's going on. I hope to recieve ...
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Question About Existence Of Branch For $(z^2 -1)^{ \frac12}$.

Consider the function $f :z\mapsto (z^2 -1)^{ \frac12}$. Now Here proved that it has branch such that $f$ is analytic in $|z|>1$.Where branch cut is $[-1,1]$. Now f is composition of two function ...
Meet Patel's user avatar
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Branch cut integral $\int_{-1}^1\left(1-x^2\right)^{\frac{1}{2}} d x$

Define the branch of $f(z)=\left(1-z^2\right)^{\frac{1}{2}}$ by the branch cut $(-\infty,-1] \cup [1,\infty), f(0)=1$. Use this branch and a suitably chosen semi-circular contour (with finite radius $...
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When working with multiple branch cuts, is there a way to chose the arguments so that log of a product can be opened as sum of individual logarithms?

Suppose a function $\eta (z)=log(\psi (z))$ where $$\psi (z)=\prod_{k=1}^{n} \left(z-z_k\right)$$ We know that $log(z)=log|z|+i(argz)$, this implies that $$log(\prod_{k=1}^{n} \left(z-z_k\right))=\...
Madhav Asthana's user avatar
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Solving complex equation, multiple solutions

I was tying to solve (1) $\sqrt{x} + \sqrt{-x} = 1$ $\sqrt(x)(1+i) = 1 \rightarrow \sqrt(x) = \frac{1}{1+i} \rightarrow x=-\frac{1}{2}i$ One can see that by symmetry $x=\frac{1}{2}i$ is also a ...
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Are both extreme values of ${\rm Arg}(z)$, $(-\pi, \pi)$, excluded in the principal branch of the complex log function?

I am confused about the definition of the principal branch of the complex log function presented in Brown and Churchill's Complex Variables and Applications (ninth edition). When we are introduced to ...
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How pathological can a Laplace transform be?

Almost every treatment of the Laplace transform that I come across talks about "the poles" of the Laplace transform function $F(s)$, thereby seeming to implicitly assume that $F(s)$ is ...
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What is the branch cut of composite of multivalued complex function

I have the following function where I want to identify the Riemann surface. $$ f(z)=\log\left(\sqrt{z^2+1}\right). \quad\quad\quad (1) $$ The square root function has a Riemann surface $R_{SR}$ with ...
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Considerations for branches of the square root on the complex plane: a down-to-earth approach

Disclaimer: I'm not a native English speaker and I'm not familiar with the English terms for mathematical objects. I will gladly edit my post based on suggestions concerning grammar mistakes and the ...
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Limit of $\log(z^2-4z+3)$ with $\text{arg}\in (0,2\pi]$

I wanted to figure out whether the solution to the following exercise is wrong: If $f(z)=\log(\eta(z))$ with $\eta(z)=z^2-4z+3$ and $\text{arg }\eta\in (0,2\pi]$, what is $$\lim_{\varepsilon\to 0^+}f(...
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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 ...
Frederic Thomas's user avatar
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Branch cut integral calculation by the method of residues.

An exercise I'm doing asks me to verify the following identity: $$ \int_0^{\infty}\frac{x^a}{\left(x+i\right)^2\left(x-i\right)^2}dx=\frac{\left(1-a\right)\pi}{4\cos\left(\frac{a\pi}{2}\right)},\quad -...
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Integrating along the closed contour crossing the branch cut

I would like to evaluate the following simple integral with complicated contour. $$ \lim_{r\rightarrow 0} \oint_{C (r)}dz~F(z) ,\quad F(z):=\left(\sqrt{z+m}+\sqrt{z-m}\right), \quad m>0 $$ where ...
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Branch cut of $\log(z^2-1)$ if $\arg(z^2-1)\in\left(-\frac{7\pi}{4}, \frac{\pi}{4}\right]$

Using the principal branch $\arg(\xi(z))\in\left(-\dfrac{7\pi}{4}, \dfrac{\pi}{4}\right]$ for $$f(z)=\log(z^2-1)=\log(\xi(z)),$$ what's its branch cut's equation/how does it look like? I was doing the ...
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Branch cuts of $\log(z^2-1)$ for unusual principal values

Say we got a multivalued function $$\log(z^2-1)\equiv\log(\xi(z)).$$ Usually, we would choose principal values such as $\arg\xi(z)\in[0,2\pi)$ or $\arg\xi(z)\in[-\pi,\pi)$. In order to find the branch ...
Conreu's user avatar
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Being precise about contour integrals when dealing with branch-cuts and singularities

In this paper https://www.biodiversitylibrary.org/item/93357#page/65/mode/1up of Landau he wants to evaluate $$\int _{2-iT}^{2+iT}\frac {x^s\log (s-1)}{s^2}\hspace {10mm}(1)$$ (page 755). He sets up a ...
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Analytic continuation of incomplete beta function

The incomplete beta function $B_x(a,b)$ is defined for $x\in [0,1]$ by the integral $$B_x(a,b)=\int_0^x dt t^{a-1}(1-t)^{b-1}.\tag{1}$$ I'm interested in two aspects associated to that. First is the ...
Gold's user avatar
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why including the origin in a closed path causes discontinuity in complex functions like the square root?

I am taking a complex analysis course and the professor confused me on the following matter: On one hand, given a complex function like $f(z)=z^{1/2}$, when we complete a closed circuit around the ...
R24698's user avatar
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A simple issue thats been swept under the rug.

Consider the integral: $$\int_0^\infty\frac{dx}{1+x^n}$$ where $n \gt 1 $ for the integral to exist. I've been trying to use contour integration to solve this and with a nice pizza slice shaped ...
Bilge K. Aksebzeci's user avatar
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Calculating the difference between $f(x+iy)$ and $f(x-iy)$ when $y\to0+$

I have asked the question about the limit of two functions as they're approaching the real axis before. Today I considered the function $f(z)=\sqrt{1-z^2}$, trying to calculate the difference in ...
Cunyi Nan's user avatar
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Calculate the integral with branches

I need to calculate the following integral $$\int_{-1}^1\frac{(1+z)^{1/4}(1-z)^{3/4}}{1+z}\, dz$$ I know how to do it by real-analysis method but I am supposed to use Cauchy theorem. But the residues ...
Yestove's user avatar
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Double complex branch cut integration, how to?

I'm working on a small project of mine, but I have gotten stuck on a particularly challenging integral: $$\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^2+1}\sqrt{x^2+ax+b}}$$ With the condition that $a^2-...
ChangedMyName's user avatar
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1 answer
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Understanding the function $f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$

I have the following complex function: $$f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$$ where $x_0 \in \mathbb R$ and $x_0 > 0$. $\kappa$ is a parameter for which we look at three cases: case: $...
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Branches of $\sqrt{z}$ when removing the negative real axis

In Complex Analysis by Ahlfors, he defines the complex square root $w(z)=\sqrt{z}$ on the complement of the negative axis $\Re(z)\leqslant 0$ to be the square root with positive real part. He then ...
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clarification on Riemann surfaces and branch cuts

This semester I am going to take Complex Analysis. I started watching the lectures, and the professor said that branch cuts and Riemann surfaces solve the problem of multivaluedness of complex ...
R24698's user avatar
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Branch cuts for $\sqrt{z(z-1)}$

I am studying for quals and found this question in a previous exam: Find a branch cut for $\sqrt{z(z-1)}$ that is analytic in $\mathbb{C} \> \setminus [0, 1]$ and takes the value of $-\sqrt{2}$ at ...
Jabbath's user avatar
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Construct a branch of $\sqrt{\log{z}}$ on the upper half-plane ${z\in C:Im(z)\geq0}$. Justify your answer

I can't quite figure out where I've gone wrong, my understanding is it should be something like: $$\sqrt{\ln\vert z\vert+iArg(z)}$$ where $Arg(z)$ is the principle argument for z.
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Branch cut of a Hypergeometric Function

Equation (9.6.8) in Lebedev (Special Functions and Their Applications) reads $F\left(\alpha ,\beta ;\frac{1}{2} (\alpha +\beta +1);z\right)=F\left(\frac{\alpha }{2},\frac{\beta }{2};\frac{1}{...
Joel Storch's user avatar
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Seemingly paradoxical logarithm manipulations inside strange gaussian like double integral

I've been playing around with an integral that seems to give two contradictory results and I cannot figure out why. Let $a$ be some real positive constant. Then on the one hand, we can consider the ...
sillyQsman's user avatar
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Stuck on (simple looking!) discontinuous double integral with gaussian weight and double pole

The question is what is the analytic answer to the limit of the difference of integrals $(I_{+\epsilon}-I_{-\epsilon})|_{\epsilon \rightarrow 0} =\int_{0}^{\infty}dx \int_{0}^{\infty}dy\left(\frac{e^{-...
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Writing $(-z)^\alpha$ in terms of $z^\alpha$

I have a very silly confusion in complex analysis. Let $z\in \mathbb{C}$ and $\alpha\in \mathbb{C}$. Now consider $(-z)^\alpha$ and suppose that we want to relate this to $z^\alpha$. One thing that I ...
user3115001's user avatar
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Evaluating $\oint\frac{dw}{(w-w_+)(w-w_-)} $ over the unit circle, where $w_\pm=z\pm\sqrt{z^2-1}$ and $z\in\Bbb{C}$ with $|w_+|\neq|w_-|$

The following problem arises in E. N. Economou's ''Green's Functions in Quantum Physics'', 3e. Evaluate the following integral over the unit circle $$ \oint\frac{dw}{(w-w_+)(w-w_-)} $$ where $w_\pm = ...
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Cuts and discontinuities of multivalued complex functions with several variables

Can somebody recommend some literature regarding cuts and discontinuities of multivalued complex functions with several variables? Thanks!
syphracos's user avatar
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Making sense of rational powers of the $j$-invariant

I have seen in the literature solutions to modular differential equations which include rational powers of the $j$-invariant (also called $j$-function), e.g., $j(\tau)^{1/p}$ for some $p \in \mathbb Z ...
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Complex analysis, Ian Stewart Exercise 4.7.5: Proving $\sqrt{z}$ is continuous on $\mathbb{C}\setminus\{x\leq0\}$

This is exercise 4.7.5 in Ian Stewart's "Complex Analysis (The Hitch Hiker’s Guide to the Plane)": Let $C_{\pi} =\{z\in\mathbb{C}:z\neq x\in\mathbb{R},x\leq0\}$ be the 'cut plane' with the ...
HIH's user avatar
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Is the textbook WRONG on the branch cut?

As defined in the textbook, it takes the branch cut along positive real axis. So the argument angle is $[0,2\pi)$. This is no problem. But in the example, it sets $f(z)=\frac1{\sqrt{2-z}}$ and claims ...
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On the meaning of multiple-valued functions

I have a very weak understanding of what does it mean for a function to be multiple-valued. We have: $$z^{\frac{1}{2}}=|z|^{\frac{1}{2}}[\cos(\theta/2+k\pi)+i\sin(\theta/2+k\pi)]$$ for $k=0,1$. We ...
davise's user avatar
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Find the analiticity domain of a complex function

I need to study the domain of analyticity of this function: $$ f(z) = \frac{\sqrt{(z-3)(z^2-4)}}{2z^2}\sin z$$ and compute the integral over the unitary circle $\gamma: \theta \to e^{i\theta}, \theta \...
Claudio Menchinelli's user avatar
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1 answer
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Approaching the branch cut and the limit of $f(x+iy)$ when $y\to0+$

Consider complex function $f(z)=\sqrt{z(z-1)}$, here $z\neq0,1$. I referred to similar questions, found branch cut and branch points. The branch points are $z=0,1$. According to the definition of ...
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Why does this show Log can't be extended to whole $\mathbb{C}^*$

Why does the following show Log can't be extended to whole $\mathbb{C}^*$? Here's another proof which I think I understand, though I'm not sure what's the connection between the two proofs: I ...
HIH's user avatar
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2 answers
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moving the branch cut in an inverse Laplace transform

I would like to calculate the inverse Laplace transform of the function $e^{-\sqrt s}$. I understand that this is given by $$ {\cal L}^{-1} [e^{-\sqrt s}] = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} ...
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