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

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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 $w\mapsto \sqrt{z}$ ...
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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 ...
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36 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 > ...
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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 ...
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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 ...
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57 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} ...
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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 $z$ ...
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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| + ...
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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 ...
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1answer
27 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 ...
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117 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 ...
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21 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 ...
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1answer
17 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 ...
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53 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 ...
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196 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 ( ...
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2answers
79 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 ...
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19 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 ...
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1answer
85 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 ...
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52 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 ...
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38 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 ...
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82 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 ...
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43 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 ...
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23 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 ...
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32 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 ...
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52 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 ...
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45 views

Find the branch function for $z^{\frac{1}{2}}$

Find the branch function for $z^{\frac{1}{2}}$ and using Cauchy-Riemann equation to prove that the branch functions are analytic for all z in Complex except along the branch cut. I did $z = ...
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22 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 ...
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122 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 ...
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54 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 ...
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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 ...
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22 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 ...
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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 ...
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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. ...
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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 ...
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41 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 ...
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77 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 ...
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40 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 ...
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Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
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Branch points of angular function

I am having difficulties finding out the branch points of the funtion $F(b^2)$ defined by $$F(b^2) = \int^{2M}_{r'} \frac{dr}{r^2\left[1-\frac{b^2}{r^2}\left(1-\frac{2M}{r}\right)\right]^{1/2}}$$ ...
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Find branches F, G of log(z), both analytic at 1 and i, such that: Im F(1) > Im F(i) Im G(1) > Im G(i)

Find branches F, G of log(z), both analytic at 1 and i, such that: Im F(1) > Im F(i) Im G(1) > Im G(i) I think I know how to do it, I'm just little confused about what "branches" mean. When it ...
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51 views

Contour integral with a different contour

This question is from the post: Contour integral with branch cut. My question is: if we choose the key hole contour with branch cut on the positive x axis, it seems that we have an addtional term: ...
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90 views

Why can't branch cut pass through poles?

In the wiki article, Example (IV) – branch cuts. Why can't we can't we choose the contour so that the branch cut is on the negative x axis. If we choose this, the two residual is out of the contour, ...
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102 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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220 views

What is a branch point?

I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a ...
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104 views

Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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67 views

Using Multiple Branch Cuts in a Contour Integral

I have the integral $$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$ which I am trying to integrate using complex integration. I know that ...
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Complex Line integral of 1/z over the principle branch cut

I would appreciate it if someone checked my work to ensure that it's consistent. Compute the integral $\int_{C}\frac 1 z {dz}$ by obtaining an appropriate branch of the logarithm. There's an ...
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49 views

Integrate function with 2 branch points

Every example I see in textbooks so far has not shown me cases like this, so please help with the following question. I wish to integrate a function $f(z)$ around the contour shown below. $f(z)$ has ...
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74 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
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146 views

Regarding branch cuts and contour integration

I am trying to compute the following integral through the use contour integration. $$ \int_0^1 \frac{dx}{\sqrt{x^2-1}} $$ So, I am considering the same integrand but from $-1$ to $1$, then doing the ...