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20 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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1answer
69 views
+50

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
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20 views

Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2]$

Problem: Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2].$ My approach to this problem was to prove that ...
3
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1answer
71 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
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0answers
22 views

How to deal with the poles on the branch cut

(1) I have a function $F(s)=\dfrac{1}{ (s+p/2)\sqrt{s(s+2p)} }$. The branch cut runs from $s=0$ to $s=-2p$. My questions is if I can consider $s=-p/2$ as a pole. If somebody can provide me with some ...
0
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1answer
58 views

Branch points and Branch cut of $f(z) = \log(z^2)$

I am trying to find the branch points and choose a branch cut for the function $f(z) = \log(z^2)$. I know that both $z = 0$ and $z = \infty$ are branch points, so it seems reasonable to just choose ...
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1answer
47 views

Finding Branch Values

I have a function $f(z) = (\frac{8}{7}z^3-\frac{64}{7})^\frac{1}{3}$ which I have found to have branch points at $z_1 = 2$, $z_2 = 2\exp(\frac{2i\pi}{3})$, and $z_3 = 2\exp(\frac{-2i\pi}{3})$. The ...
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0answers
21 views

Complex integral with branch cuts.

The problem is the following, $$ \ \int_{-\infty}^{\infty} du e^{-iu w }\bigg( \cos (\theta u) - i \frac{\xi }{\theta} \sin (\theta u)\bigg)^{-1/2} $$ When we go to complex $u$ plane there are branch ...
5
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4answers
225 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
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38 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
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0answers
28 views

Suitable branch for Arg

I have the following function: $ \ f(z)=\log(z^3-2) $ and I am asked to find a branch where $f(z) $ is defined. I understand that since $ f(0)=\log(-2)$ the main branch of Arg is not defined and we ...
2
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0answers
116 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
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2answers
61 views

Defining a branch of $(1-\zeta^2)^{-1/2}$

In this question I brought up a passage from Stein/Shakarchi's Complex Analysis page 232: ...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z \frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the ...
2
votes
3answers
78 views

power series expansion of $z^a$ at $z = 1$

I'm working through some problems in a complex analysis book, and one asks to compute the power series expansion of $f(z) = z^a$ at $z = 1$, where $a$ is a positive real number. The series should ...
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2answers
92 views

Choose the branch for $(1-\zeta^2)^{1/2}$ that makes it holomorphic in the upper half-plane and positive when $-1<\zeta<1$

From Stein/Shakarchi's Complex Analysis page 232: ...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z \frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the integral is taken from $0$ to $z$ along ...
5
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2answers
119 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
3
votes
1answer
85 views

Branch cut problem

I am looking at the text by G. K. Batchelor, An Introduction to Fluid Dynamics, pg. 428-9. I am looking at the inverse mapping of $z = \zeta + \frac{\lambda^2}{\zeta}$ given by ...
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1answer
89 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
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1answer
39 views

Holomorphic problem

I have a function $f(z)$ holomorphic in $\mathbb{C}\setminus\mathbb{R}^-$. I have these information: $f(x+i\epsilon) = f(x-i\epsilon)$ on $\mathbb{R}^+$ (the $\epsilon$ is indented as a shorthand ...
0
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0answers
110 views

Difficult Fourier transform

While looking at non-local modifications to wave propagation in 2d I have run into the following integral $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega dk \ln(k^2-\omega^2)e^{-i\omega ...
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42 views

Why is the Chern class of a line bundle well defined?

Let $M$ be a compact Riemann surface with a finite open covering $$ M = \bigcup_{i=1}^{n} U_i $$ which has the property that every intersection is contractible (i.e. it is a good cover). To each two ...
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1answer
102 views

Branch cut for $\log (iz)$ in the region $\{z:\mathrm{Im}(z)>0\}$

If someone could explain branch cuts and branch points to me that would be fantastic. I understand that a branch cut is a curve that we remove from the domain to make a function (usually a logarithm) ...
0
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1answer
78 views

Branch cut of $w=\left(\dfrac{z+1}{z-1}\right)^{1/3}$

I'd like todetermine a branch cut for the function $w=\left(\dfrac{z+1}{z-1}\right)^{1/3}$ that allows to construct analytic branches defined on $|z|>1 \;,\; \forall z\in \mathbb C$. How can I do ...
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0answers
91 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
0
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0answers
70 views

Branch cuts and multidimensional residues

When dealing with a multidimensional contour integral, how do you treat multivalued functions with branch cuts? In simple domains with singe valued functions it's not too hard to calculate these ...
0
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2answers
701 views

Finding branch points and branch cuts of arctan

I am studying complex analysis and I do not yet fully understand branch points and branch cuts. I am trying to figure out how it works by looking at the following: $z \rightarrow \frac{1}{2i} ...
1
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0answers
24 views

possible results of integral along closed path after defining branch of sqrt

Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ ...
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1answer
60 views

edge disjoin Cut Set

prove that a graph G=(V,E) where | v | =n there are at most n-1 edge disjoint cut sets. I was thinking that for tree it is true since each edge is cut set. but i have no idea how to prove above ...
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1answer
95 views

handwaving substitution in integral involving branch cut and derivative of sqrt + generalization

Want to compute $$ I = \int_0^i \mathrm{d}z \frac{z}{\sqrt{z^2-1}}$$ on the complex plane using complex methods. QUESTION: is the result $i \left( \sqrt{2}-1 \right)$ which one gets imposing ...
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0answers
215 views

Branch Cut for $\ln(1 - z^2)$

"Given that $g(z) = \ln(1-z^2)$, defined on $\mathbb{C}\backslash \left(-\infty, 1\right]$, i.e. the branch cut is from $-\infty$ to $1$ along the real axis. Find $g(-i)$ given $g(i) = \ln(2)$" I ...
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2answers
108 views

Is discontinuity along a line equivalent to branch cut?

Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this? Is it sufficient to prove that $f(z)$ is discontinuous across this line? ...
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2answers
93 views

Help with complex logarithms

For real $x$ what does $-\ln(1-e^{2\pi i x})$ equal so that it agrees with the series expansion, how would I find the real and imaginary parts. $$-\ln(1-e^{2\pi ix})=\sum_{n=1}^\infty\frac{e^{2\pi i n ...
0
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1answer
80 views

contour integral with integration by parts

Is there a complex version of integration-by-part? I saw someone used it but didn't find it in textbook. I tested integrals $\int_{\mathcal{C}}\frac{\log(x+1)}{x-2}\mathrm{d}x$ and ...
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0answers
184 views

Need help to understand branch cuts

I have a question about branch cuts. Suppose you have $f(z) = \sqrt{z^2 -1} $. Then the branch points are $ \pm 1$, so we can make a branch cut from $ (- \infty , 1]$ in order to define $f$ ...
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0answers
60 views

$\log\Gamma (z)$ vs $\log(\Gamma(z))$: formula to have the correct branch cut?

The complex logarithm $\log$ is a multivalued function: consequently some branch cut choices are necessary. Most softwares introduce a single branch cut along the negative real axis. Moreover, when ...
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2answers
1k views

Contour integral of $\sqrt{z^2-1}$ on $|z| = 2$

I've been wrestling with this problem (available here ): Evaluate the integral of $f(z) = \sqrt{z^2-1}$ around the circle $\{z: |z| = 2\}$, where the branch of the square root function is chosen so ...
3
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1answer
139 views

A branch cut problem

In Ahlfors' Complex Analysis text, chapter 3, section 4 the transformation $z=\zeta+\frac{1}{\zeta}$ is discussed. The author notes that for every $z$, there exists 2 solutions for $\zeta$ and they ...
5
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2answers
404 views

Calculating integral with branch cut.

I'm learning how to calculate integrals with branch points using branch cut. For example: $$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$ where ...
2
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2answers
257 views

difficulty understanding branch of the logarithm

Here is one past qual question, Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$ (Hint: start by defining an appropriate ...
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1answer
152 views

Square root principle value convention

Why is the principal square root of a complex number defined as $\sqrt z = \sqrt r e^{-i \varphi / 2}$ for $\varphi \in (-\pi, \pi]$ ? Wouldn't it be more natural to let $\varphi \in [0, 2\pi)$ as it ...
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4answers
339 views

Proving $\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$.

I am trying to prove $f(z)=\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$. The issue is proving there is no branch point. If I try the approach of taking the path $z=1+r\exp(i\theta)$ with $r=\epsilon$ ...
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1answer
119 views

How do we know how many branches the inverse function of an elementary function has?

How do we know how many branches the inverse function of an elementary function has ? For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
5
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2answers
2k views

Branch cut for $\sqrt{1-z^2}$ - Can I use the branch cut of $\sqrt{z}$?

I was trying to clarify some questions I had about elliptic integrals using http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf There they define the map $$\phi\colon w\mapsto ...
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3answers
122 views

Improper integration using complex methods

Sorry for my English if there is any mistake. The exercice for which I need help is the following: Compute using complex methods: $I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$ i) Choose the complex ...
3
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1answer
975 views

Complex Analysis - Question about branch cuts

I am having trouble understanding how branch cuts work. For example, the function $f(z)= \sqrt{z}$ has a branch cut where you reject the negative real axis. But how do you define the output so that ...
2
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1answer
240 views

Analytic branches of $z^{-i}$.

How to describe all the branches of the function $z^{-i}$, analytic in the whole complex plane except the positive real axis? I consider $z^{-i}=e^{-i \log z}$ and the branch becomes whole complex ...
3
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1answer
346 views

summation of an infinite series involving arctan

I'm having problems with the following calculation. Let $a >0$ $$ \begin{align} & \sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) = \text{Im} \sum_{n=1}^\infty \log \left( 1 + ...
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1answer
157 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...