# Questions tagged [branch-cuts]

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

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### Understanding where branch cuts live

I understand how we define $e^z = e^x e^{iy}$ and use this to define the multi-valued function $\log(z) = \ln(r) + i(\theta + 2\pi n)$. I think I also understand how we may take any branch by setting ...
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### Branch points: Find value of $f(z)=\sqrt{\pi^2 + \log^2(z)}$ at $f(i)$

I have the function (and the problem) mentioned in the title and I am given the initial condition of $f(1)=\pi$. The cut is a rather strange one: It is the portion of the unit circle that connects $-1$...
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I'm trying to understand branch cuts in complex analysis. Let's say I have the following function $$f(z)=(z^3+1)^{1/3}.$$ I see that the branch points in this case are the solutions to the equation $z^... 0 votes 1 answer 50 views ### Find$\operatorname{Log}(1-i)$where$\operatorname{Log}z$is the principal value. My attempt starts with$\log z=\operatorname{Log}|z| + i\operatorname{Arg}z$. I start with the$\operatorname{Arg}$. We know $$|1-i|=\sqrt{1^2+1^2}=\sqrt{2}\implies\sin\theta=-\frac{\sqrt{2}}{2}\... 1 vote 0 answers 16 views ### Integral Representation for the Legendre Function of the First Kind In the text Special Functions and their Applications by N.N. Lebedev (pp.172-173), a derivation is presented for a certain integral representation of the Legendre function of the first kind P_{\nu }(\... 2 votes 2 answers 65 views ### Evaluating \int_{1/\sqrt{2}}^1\frac{1}{(3y^2-1)\sqrt{2y^2-1}}\,\mathrm{d}y; how do you avoid using a complex substitution? \newcommand{\d}{\mathrm{d}}The given exam question - I provide the beginning for context: Let:$$I=\int\frac{1}{(b^2-y^2)\sqrt{c^2-y^2}}\d y$$Where b,c\gt0, and employ the substitution y=\frac{... 5 votes 2 answers 184 views ### Evaluate \int_{-\infty}^{\infty}\frac{\sinh\left(y\sqrt{\alpha^2-\omega^2}\right)}{\sinh\left(H\sqrt{\alpha^2-\omega^2}\right)}e^{i\alpha x}d\alpha I need to evaluate the Fourier inverse integral \displaystyle \int_{-\infty}^{\infty}\frac{\sinh\left(y\sqrt{\alpha^2-\omega^2}\right)}{\sinh\left(H\sqrt{\alpha^2-\omega^2}\right)}e^{i\alpha x}d\... 0 votes 0 answers 22 views ### Removable branch point discontinuity? While reading this paper https://arxiv.org/pdf/1105.3426.pdf on Fourier Extensions, I came across something I can't get my head around. Here is the thing: Suppose you have an entire function f : \... 0 votes 1 answer 29 views ### analytically continuing a function into a two-leaf function My function is defined as$$f(z) = \int_0^\infty \frac{\rho(x) dx}{x-z } , $$where \rho(x) is a real function. It is easy to see that f has a branch cut along the positive real axis. Now let us ... 0 votes 0 answers 24 views ### Prove or disprove: analytical function Prove or disprove: There exist an analytical function f(z) in a deleted neighbourhood of z=0 such that f^2(z) = z. I disproved it with the following argument: the function f(z) is given by ... 0 votes 0 answers 18 views ### Integration along a contour containing a branch cut Consider the following integral I = \int_{-\sqrt{b^{2} + a^{2}}}^{\sqrt{b^{2}+a^{2}}} (a^{2} - z^{2})^{\lambda}e^{- i \omega z}\mathrm{d}z In this integral, a, b and \omega are real numbers. ... 0 votes 1 answer 45 views ### Determine a branch of f(z)=\log(2iz−z^2) Determine a branch of f(z)=\log(2iz-z^2) that is analytic at z=1. Then find f(1) and f'(1). First we note that g(z)=2iz-z^2 and recall \mathcal{L}_\tau:=\log|z|+i\arg_\tau z. So a branch ... 1 vote 1 answer 96 views ### z/\sqrt{-z^2}=-i when \operatorname{arg}(z)\leq 0? Looking at asymptotic expansions for the imaginary error function I find the following for z\in\Bbb C\setminus\{0\}:$$ \tag{1} \frac{z}{\sqrt{-z^2}}= \begin{cases} -i, &\operatorname{arg}(z)\... 0 votes 2 answers 63 views ### Which branch of the square root allows$(z^2)^{1/2} = z$I am aware that$(z^2)^{1/2} = z$does not hold for every branch of the function$f(z) = z^{1/2}$. For example if we do not take a branch and consider$z=-1$, we get $$f((-1)^2) = \exp(\frac{1}{2}Log((... 0 votes 0 answers 35 views ### How to find a branch which is analytic on the exterior of the unit circle for \sqrt(z^2 +1), |z| > 1 I know we can rewrite \sqrt{z^2 +1} as z^2 (1+z^{-2}) and use this by looking at the principal branch of the function \exp{\left(\frac{1}{2} \log(1+z^{-2})\right)}. However I am struggling to ... 3 votes 1 answer 102 views ### Difficulty evaluating \sum_{n=1}^\infty\frac{1}{n^2}\sqrt{\alpha n-1} for a real \alpha\gt1 \newcommand{\d}{\,\mathrm{d}}I've done my best with this series, but I've never actually seen a sum of square roots before! The following question was given to me by a friend - apparently some ... 1 vote 0 answers 52 views ### How to find branch cuts and branch points of \sqrt(z^a +1) I have tried to find out the branch cuts and branch points of the function \sqrt(z^a +1), where 0<a<2. The function can be written as e^{\frac{1}{2}(\log(z^a+1))} and from here I have ... 2 votes 2 answers 85 views ### Finding branch cuts Let \omega=-\frac{1}{2}+\frac{\sqrt{3}}{2}i. I would like to find branch cuts so that the complex function$$f(z)=\sqrt{z(z+1)(z-\omega)}$$can be defined continuously off the branch cuts. I ... 0 votes 1 answer 74 views ### Confusion about the change of variable z \to \frac{1}{z} for a multivalued function I'm currently struggling with something that came up in my studies. I'm trying to integrate a multivalued function like the square root on a given path, specifically a function with two branch points, ... 1 vote 0 answers 51 views ### How to improve the solution on this problem of branch cut I was given the following problem. Consider the function w=(z^2-4)^{1/2}. Insert branch cuts in the z-plane such that \pi/4>\arg{w}>-3\pi/4. Find the equations of the branch cuts in the z-... 0 votes 0 answers 48 views ### Find branch cut of w=(z^2-4)^{1/2} such that \pi/4>\arg{w}>-3\pi/4 I was given the following problem. Consider the function (z^2-4)^{1/2}. Insert branch cuts in the z-plane such that \pi/4>\arg{w}>-3\pi/4. Find the equations of the branch-cuts in the z-... 1 vote 1 answer 36 views ### Branch cut contour integral - contour understanding [closed] Im trying to understand the contour when there are two poles on the real axis without poles on Im axis, because their residue cancle each other. contour$$I=\int_0^\infty\frac{dx}{(x+1)^2(x+2)}$$For ... 3 votes 1 answer 58 views ### Deducing \int_0^{\pi}\log \sin x dx =-\pi\log 2 from \int_0^{\pi}\log (-2ie^{ix}\sin x) dx = 0 On Ahlfors Complex Analysis, chpter 5.3, the author explains how to evaluate \int_0^{\pi}\log \sin x dx =-\pi\log 2. He does so by using Cauchy's formula to deduce that$$\int_0^{\pi}\log (-2ie^{ix}\... 1 vote 0 answers 71 views ### Square root of continuous function on simply connected domain It is well that every holomorphic function on a simply connected domain which is everywhere non-zero has a holomorphic square root. But is it true that every continuous function on a simply connected ... 0 votes 0 answers 65 views ### Square root branch selection of a polynomial The complex function $$f(z) = \sqrt{(z-(a+b)) (z-(b-a))(z-(a-b)) (z+(a+b))}, \quad a,b\in \mathbb{R}, \quad a,b >0, \quad a > b$$ has 2 branch cuts located between$z \in [-a - b,b - a ]$and$...
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 ...