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

learn more… | top users | synonyms

2
votes
2answers
79 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log z}{z}...
2
votes
2answers
86 views

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
1
vote
1answer
62 views

Integrating $z^i$ over the unit circle

Why is there a difference between integrating it over a unit circle parametrized over $t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $t \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right]$?
2
votes
0answers
17 views

Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
3
votes
2answers
91 views

Contour integral of $\int^\infty_0 \frac{x^{n-1}}{1+x}dx$

Prove $$\int^\infty_0 \frac{x^{n-1}}{1+x}dx= \frac{\pi}{\sin(n\pi)}$$ I would like to consider two circles of radii $\epsilon < 1 < R$ centered at the origin and connected by a corridor of ...
0
votes
0answers
22 views

Alternative way of finding a logarithm branch of a polynomial in a certain set

First the question: Let $$r(z) = a_n + a_{n-1}z^{-1} + ... + a_0z^{-n}$$ I need to show that there exists a $K>0$ so that there exists a branch of $log(r(z))$ in $D = \{ |z| > K \}$. I know ...
0
votes
0answers
16 views

Analytic Branches

How would I show that there exists an analytic branch of $$(1-cosz^2)^\frac{1}{4}$$ near $z=0$? My initial thoughts is to use power series expansion since, the expansion of $cos z $ converges for all $...
3
votes
1answer
53 views

How to do this infinite sum?

$$ F_\alpha(z)=\frac{2}{\pi \alpha}\sum_{n=0}^\infty \frac{1}{n^2+z/\alpha^2} $$ The answer is: $$ F_\alpha(z)=z^{-\frac{1}{2}}\coth(\pi z^{\frac{1}{2}}/\alpha)+\alpha/\pi z $$ It is easy to show ...
-2
votes
1answer
38 views

Integrating along a branch cut

What contour would one use to integrate the following equation? $\int_{0}^{\infty}\frac{x^a}{(x^2+1)^2}dx$ where $-1 < a <3 $ and $x^a= e^{aln(x)}$
3
votes
1answer
32 views

How is the principal branch of logarithm defined?

In my textbook, it is defined as: $$\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$$ Where $\operatorname{Arg}$ is the principal branch of $\arg$, that's, the function which outputs the ...
0
votes
1answer
31 views

Writing $1/(X+i\epsilon)$ in terms of principal value and imaginary part

I am looking to derive the relation $$\frac{1}{X + i\delta} = \text{P.V} \frac{1}{X} - i \pi \delta(X)$$ In particular, I don't see where the factor of $\pi$ comes from in the derivation. I proceed by ...
1
vote
1answer
34 views

Branch cuts to infinity

The function $$f(z)=\sqrt{z^2-1}$$ does not have a branch point at infinity. However, people often take branch cuts going from $z=\pm 1$ to $\infty$ along the real positive and negative axis ...
3
votes
2answers
84 views

Integrate $\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$

I'm trying to evaluate the integral below for my research related to sound radiation. Assume $a$ is a positive constant. $$\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$$ First,...
2
votes
1answer
47 views

Continuity of principal complex arccos

The principal branch of the complex arccos is defined by $$\operatorname{Arccos} z = -i \operatorname{Log}\left(z + i\sqrt{1-z^2}\right)$$ where $\sqrt{\cdot}$ and Log denote the principal branches of ...
0
votes
0answers
17 views

Limit argument for complex squareroot

Let $z\in \mathbb{H}\backslash(0,i]$, where $\mathbb{H}$ is the upper half plane. I want to show that $z(\sqrt{z^2+1}-z)\rightarrow \frac{1}{2}$ for $z\rightarrow \infty$, where $w\mapsto \sqrt{z}$ ...
0
votes
0answers
34 views

Branch point at infinity?

I have to find the branch points of $f(z)=\left( z(z+1)\right )^{1/3}$. It is clear that $0$ and $-1$ are branch points, but I am not sure about infinity. Making the substituition $x=\frac{1}{z}$ and ...
2
votes
1answer
47 views

express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > \frac{1}{...
0
votes
0answers
18 views

Express function in terms of polar coordinates and find residues of poles

The function $f(z)$ is given by $$f(z) = (z + \sqrt{3})^{1/2}ln(z-1).$$ The branch of this function is such that $$-\frac{4\pi}{3}<arg(z-1)\le\frac{2\pi}{3} and -\frac{\pi}{2}<arg(z+\sqrt{3})\...
1
vote
0answers
27 views

Branch cut of $e^{iz^{1/2}}$

Wolfram alpha tells me that $$ z \mapsto e^{iz^{1/2}}$$ does not have any branch cuts in the complex plane. However, I am skeptic. If this is true, can someone explain why? [Here is the wolfram ...
0
votes
0answers
58 views

A trick to make $\frac{1}{i} = i$ (contradicting $\frac{1}{i} = -i)$ [duplicate]

Bizarrely came up with this while asleep ... $$ \tag{1} \frac{1}{-1} = \frac{-1}{1} $$ and $$ \tag{2} \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$ Thus combining (1) and (2): $$ \tag{3} \sqrt{\...
3
votes
2answers
72 views

Calculating residues of function with branch cut

Show that $$I= \int_{0}^{\infty} \frac{\ln x}{x^\frac{3}{4} (1+x)} dx = -\sqrt{2} \pi^2$$ I used a keyhole contour, with branch point at $z=0.$ Around $\Gamma$, $|zf(z)|$ tends to $0$ as $z$ tends ...
3
votes
0answers
58 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
3
votes
3answers
87 views

Contour integral with a logarithm squared

The integral I'd like to evaluate is $\int_0^\infty \frac{\log^2 x \, dx}{(1+x)^2}$. I consider the function $f(z) = \frac{\text{Log}^2 z}{(1+z)^2}$, which has a pole of order 2 at $z=-1$ and has a ...
1
vote
1answer
28 views

Identity for the derivative of an arbitrary branch of $z^{1 / n}$

I'm reviewing old homework problems. Let a function $f(z)$ be some branch of $z^{1/n}$. Show that $$f'(z) = \frac{f(z)}{nz} \textbf{.}$$ I wrote: Let $z = re^{i(\theta+2\pi k)}$ and fix $k$...
6
votes
1answer
146 views

Contour integral of $\sqrt{z^{2}+a^{2}}$

Suppose $a$ is real and nonnegative. Say we wanted to compute the above function (for whatever reason, be it to solve an improper real integral, or something else) along the curve $C$, as on the ...
0
votes
1answer
29 views

Logarithm rule for branch cut logarithms

I know that for $a, b \in \mathbb{R}$, the rule $\log(ab) = \log(a) + \log(b)$ holds. What about for $a_1, b_1$ in the right half-plane, or $a_2, b_2$ in the sector from $\frac{-3\pi}{4}$ to $\frac{...
0
votes
1answer
19 views

How to Express the domain of the mult-valued function $\sqrt{1-z^{2}}$?

In the analysis of the function $w(z)=\sqrt{1-z^{2}}$ I've set the branch cuts from $-\infty$ to $-1$ and from $1$ to $\infty$. I would like to know how can I express the domain of this function when ...
1
vote
1answer
68 views

contour integral branch cut

I need some help to solve the following integral by contour integration. $$\int_{0}^{1} x^a (1-x)^{1-a}\,\mathrm{d}x$$ I attached my ideas and a picture of the paths to fix the labels. Kind regards,...
7
votes
2answers
223 views

Choosing parametrization for complex integration with two branch cuts

I am particularly interested in how Ron Gordon came up with the parametrization in his anser to this question: Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\...
2
votes
2answers
106 views

Branch cut for $\sqrt{1-z^{2}}$ and Taylor's expansion!

I'm working in a problem that involves the equation $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface ...
1
vote
0answers
23 views

Obtaining branches

I'm reading the Ahlfors book of Complex Variable, and I don't understand how to obtain one branch of a single-valued function. Can someone explain me how to do this in a way more extensive for any ...
2
votes
1answer
256 views

residue theorem with logarithmic function

I have problem integrating function with logarithm. Problems seems always to be branch cut of $\log$, but here it is different I think. I have task to integrate $$\oint_{|z| = 1} \! dz \log\left(\...
0
votes
1answer
68 views

Branch cut problem, square root

I am looking at $f(z)=\sqrt{1-z^2}$ and a branch cut on the real axis from $z=-1$ to $z=1$. Is it correct to say that $f(x+i\epsilon) = -f(x-i\epsilon)$ when $x\in(-1,1)$, $\epsilon\in\mathbf{R}$ and ...
0
votes
1answer
49 views

Branches of the logarithm function problem

Let $D$ be a domain in $ \mathbb{C} \setminus \{0\}$ such that the annulus $\{z\in \mathbb{C} : 1<|z|<2\}$ is contained in $D$. Prove that there is no branch of the logarithm defined in $D$. My ...
2
votes
1answer
88 views

How do I prove that $\int_0^1 \frac{1}{(x^2-x^3)^{1/3}} =\frac{2\pi}{\sqrt{3}}$?

This is a problem from Mathematical Methods for Physicists, by Arfken, 7th edition (Problem 11.8.27). I know the integrals in the circular paths around 0 and 1 will vanish, but am completely lost on ...
0
votes
1answer
65 views

Branch cut of the arctan, and integrating in the complex plane

I'm doing a Complex Analysis class this semester and I've got this interesting integral problem. I have to do the following integral: $$\int_\Gamma \frac{1}{1+z^2}dz$$ Where $\Gamma$ is the circle ...
1
vote
1answer
34 views

Why does the branch cut for the principal branch of log(z+1) start at z=-1?

If I cut away the negative real axis to make log(z+1) single-valued, why does the branch cut start at z=-1 and not at the origin z=0? Why does the shift in argument from log(z) to log(z+1) make it ......
0
votes
0answers
42 views

Branch Cut in Complex Antiderivatives

I am reading James Ward Brown's Complex Variables and Applications and I am stuck on this problem. Problem: Show that $$\int_{-1}^{1} z^i dz=\frac{1+e^{-\pi}}{2}(1-i),$$ where the integrand denotes ...
2
votes
1answer
59 views

Complex Analysis, the Sum of Residues is Imaginary?

I'm trying to solve this problem using complex analysis: $\int_0^\infty\frac{\sqrt{x}}{x^2-2x+3}\, dx$ I'm using a contour that avoids the branch cut along the +x axis of the complex plane, looking ...
0
votes
0answers
24 views

Find out the branch cut that lies along the imaginary axis but avoids the points $z = 2i$ and $z = -􀀀2i$

Find $ln(z + i)-ln (z-i)$ State the branch points and find out the branch cut that lies along the imaginary axis but avoids the points $z = 2i$ and $z = -􀀀2i$.For your choice of branch cut, give the ...
7
votes
1answer
126 views

How to choose the right branch to find the roots.

I want to find the roots of $$f(z)=\left[a+zg(z)\right]^2+g(z)^2=0$$ Where $a$ is real number and: $$ g(z)=\frac{1}{2\sqrt{z^2+1}}\ln\left(\frac{z+\sqrt{z^2+1}}{z-\sqrt{z^2+1}}\right) $$ It is ...
1
vote
2answers
65 views

Roots and Logarithms of Matrices.

Ok, this question may be too broad or fuzzy, if it is please let me know and I'll try and sharpen or narrow it down a bit. Hi. I am aware of some of the difficulties of defining roots and logarithms ...
8
votes
1answer
110 views

Why would a branch cut not end at a branch point?

Both Wikipedia and MathWorld (here and here) seem to place some imporantance on saying, but not elaborating on It should be noted that the endpoints of branch cuts are not necessarily branch ...
1
vote
0answers
32 views

Choose particular branch cuts for $\ln \frac{\cosh z + \sqrt{\cosh^2z - \cosh^2a}}{e^z \cosh a}$

Is there a way to define the following multivalued function as a single valued function where the branch cuts are taken to be $z =x+iy \in \{(x,y):|x|<a, \hspace{3mm}y = i\pi n \}$ and $n$ an ...
1
vote
2answers
118 views

Solving $x^{2n} = \frac{1}{2^n}$ for $x$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for $...
1
vote
0answers
22 views

Branch cut of $z^{-s}$

I need to perform the following integration: $I(s) = \int_{\gamma}\text{d}z\ z^{-s} \frac{\text{d}\ln\mathcal{F(z)}}{\text{d}z}$, where $\mathcal{F(z)}$ is analytic everywhere on the complex plane. ...
3
votes
2answers
75 views

Simplifying $\frac{1}{\sqrt{-1}}$ [duplicate]

When trying to simplify $\frac{1}{\sqrt{-1}}$, you could rationalize it: $$\frac{1}{\sqrt{-1}}\cdot\frac{\sqrt{-1}}{\sqrt{-1}}=\frac{\sqrt{-1}}{-1}=-\sqrt{-1}$$ Or you could simplify it as one radical:...
0
votes
1answer
44 views

How do I recognize branch points?

For instance, $z^2$ - 1 has branch points at i and -i, but that doesn't seem obvious at all - and writing this function using the exponential and complex logarithm functions doesn't seem to help ...
3
votes
1answer
97 views

single valued analytic branch of multivalued function

Consider $f(z)=\sqrt{z\sin z}$. Can $f(z)$ be defined near the origin as a single valued analytic function? How do we choose the branch cut. The answer is here http://math.nyu.edu/student_resources/...
2
votes
1answer
46 views

Finding branches of $z^{ab}$

Let $f: G \to \bf{C}$ and $g: G\to \bf{C}$ be branches of $z^a$ and $z^b$ respectively ($a,b\in \Bbb C$). Suppose that $f(G) \subset G$ and $g(G) \subset G$. Prove that both $f\circ g$ and $g \circ f$ ...