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Questions tagged [residue-calculus]

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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Black hole integral

What I call as ‘black hole’, has a formal name of ‘limit point of singularities’. Suppose $k$ is a black hole of the function $f$ (e.g. $0$ is the black hole of $\csc \frac1z$), then how to evaluate ...
1
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1answer
36 views

Calculate integral with $z^a$ using residue theorem

I am trying to solve the following: Find $\int_0^\infty \frac{x^a}{x^2+3x+2}dx$ for $0<a<1$ by using the residue theorem. I thought to let $f(z)=\frac{z^a}{z^2+3z+2}$ and by taking the ...
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1answer
22 views

Integration of digamma function

I was trying to perform the contour integral of the digamma function $\oint\limits_C \psi(z)\,dz$ on the neighborhood (a small circle $-k+re^{it}$, $k \in \mathbb{Z}$ ) of $k$, before actually ...
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1answer
63 views

How important is the assumption $\gamma$ is positively oriented? (Residues, Cauchy's Thm from Cauchy's Integral Formula)

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.32, Cor 8.27 Question 1. Should the following 2 statements in the textbook have an ...
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0answers
61 views

Are these the Big and Little Picard Theorems?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.3, 9.4 These seem to be the Big and Little Picard Theorems or at least related to them. ...
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2answers
111 views

Proof of Casorati-Weierstrass [on hold]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2 I have questions on the proof of Casorati-Weierstrass Theorem (Thm 9.7) - If $z_0$ is ...
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0answers
20 views

Inverse Laplace, residue, simple or essential pole from bivaluated hyperbolic trigonometric function?

I would like to compute the following function: $$f(t)=\mathcal{L}^{-1}\Big[\frac{1}{s(e^{a+\text{arcosh}(s+\cosh a)}-1)}\Big](t)$$ However, it seems that there is no other pole than the pole of ...
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1answer
100 views

Must a positively oriented path be simple, closed and piecewise smooth? [on hold]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch1, Ch4 Question 1: Do all paths have an orientation? The following is a quote from ...
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1answer
183 views

Prove $ \lim_{z \rightarrow z_o} f(z) \in \mathbb{C}$ if $\lim_{z \rightarrow z_o} (z-z_0) f(z) = 0$

Context: A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.1, 9.3 (asked about here) Question From (here): How do we have that $$\...
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1answer
60 views

Holomorphic $f$ has a pole $\iff f(z) = \frac{g(z)}{(z-z_0)^m}$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 9.2 Cor 9.6 of Prop 9.5(*) Suppose $f$ is holomorphic in $\{0<|z-z_0| < R\}$. ...
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1answer
102 views
+50

$f$ zero (essential singularity) $\implies \frac 1 f$ pole (essential singularity)

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.1, 9.3 How do I do these? (I converted attempts to an answer.) (Exer 9.1) Prove $f$ ...
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3answers
69 views

Calculating $\int_{-\infty}^{\infty}\left( \frac{\cos{\left (x \right )}}{x^{4} + 1} \right)dx$ via the Residue Theorem?

In the text, "Function Theory of One Complex Variable" by Robert E. Greene and Steven G. Krantz. I'm inquiring if my proof of $(1)$ is valid ? $\text{Proposition} \, \, \, (1) $ $$\int_{-\...
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2answers
59 views

Residue at infinity calculating integrals

I have the following problem which I want to evaluate at infinity: $$\oint \dfrac{(z+2)}{(z^2+9)}dz$$ I approach this problem by saying that $z=\dfrac{1}{t}$ and $dz=\dfrac{-1}{t^2}dt$. And I plug ...
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0answers
25 views

Integration of definite integrals with residue theorem $\int_0^{2\pi}\frac {\cos(2\theta)} {5+4\cos(\theta)}\, d\theta.$ [duplicate]

$$\int_{0}^{2\pi} \dfrac{\cos2\theta}{5+4\cos\theta}d\theta$$ I am trying to take this integral. Solution manual of the book I'm using suggesting that I should take $\oint \dfrac{z^2}{5+2(z+z^{-1})}...
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1answer
48 views

Is $\int_{C[-2i,r]} \frac{dz}{z^2+1} = 0 , 1 < r < 3$? I got $-\pi$.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.33 It is given that $r \ne 1,3$ and that the answer is $0 \ \forall r$. I got $0 \ \...
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0answers
22 views

Residue Calculus Integrable Model

I'm currently working on the derivation of Bethe ansatz equations. I have an equation like this: $ z^{-1}\prod_{m=1,m\neq l}^{Q-1}\frac{q^2z-z_{m}}{z-z_{m}}+q^{-2}z \prod_{m=1,m\neq l}^{Q-1}\frac{q^{...
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1answer
37 views

Finding simple fraction decomposition with help of Taylor's Theorem and Residues theorem

I have this problem: $F(s)=\frac{a_{1}s+a_{2}}{s^{2}(s^{2}+12s+37)}$ I thought in Taylor's expansion of a function f(s) in s=0: $f(s)= f(0)+f'(0)\cdot s+f''(0)\cdot s^{2}+\cdots$ and then I defined:...
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18 views

Loop integral around pole lying on logarithmic branch cut

I want to evaluate a general case of an integral $$\lim_{r\to0^+}\oint_{|z-q|=r} f(z)\ln(z-k)dz$$ where $f$ is meromorphic on the whole complex plane and $q$ is a pole/essential singularity of $f$. $...
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1answer
30 views

What happens if I have an essential singularity and a pole for the same $z$?

for instance $$\dfrac{\sin(\dfrac{1}{z})}{z}$$ $z=0$ is a pole for the denominator but $z=0$ is an essential singularity for the numerator too. So how does it work ? i have two residues ? or it's ...
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1answer
32 views

Residue on all layers of a complex function

I understand how residues work for single-layered functions, but one question type from my assignment has me puzzled. Say, $$f(z)={\sqrt{z}\over{(z-z_0)}}$$ has a simple pole at $z=z_0$ and its ...
6
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1answer
124 views

Variation of residue theorem?

Residue theorem can be stated informally as $$\oint_C f(z)dz=2\pi i\sum a_{-1}$$ A contour integral sums up all the $-1$ coefficients inside. Then, one would naturally ask: Is there something ...
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1answer
41 views

Find the residue of the function $f(z) = \frac{z^3}{(z-1)(z^4+2)}$ at $z=0$

Problem: Calculate the residue of the function $f(z) = \frac{z^3}{(z-1)(z^4+2)}$ at $z=0$ I am confused on where to even start on this question since there is no pole at z=0. Is z=0 even a ...
3
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2answers
92 views

A Strange Mistake in Application of Residue theorem $\int_0^{2\pi}\frac {\cos(2\theta)} {5+4\cos(\theta)}\, d\theta.$ [closed]

$$\int_0^{2\pi}\frac {\cos(2\theta)} {5+4\cos(\theta)}\, d\theta.$$ While applying the calculus of residues to the above problem I'm getting the answer as $ 19\pi/24$. I have tried many times and ...
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1answer
28 views

Complex integral containing Gaussian

I want to calculate the integral using complex integration: $$ f(t) = \int_{-\infty}^\infty \dfrac{e^{-(z+iat)^2}}{4z^2+1} dz = \int_{-\infty}^\infty \dfrac{e^{-(z+iat)^2}}{(2z-i)(2z+i)} dz$$ where $a$...
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1answer
53 views

Prove $\frac{1}{2 \pi i}\int_{\gamma} \frac{gf'}{f} = \sum_{m=1}^{j} g(z_m) - \sum_{n=1}^{k} g(p_n)$.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Pf: By Thm 8.14 and Cor 9.6, $\exists$ holomorphic $h: G \to \mathbb C$ s.t. $h(z) \ne 0$ ...
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1answer
89 views

Is $\int \frac{e^{z^2}}{z^3} dz=\pi i$? I got $2 \pi i$.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka The answer key says $\pi i$. By Cauchy's Integral Formula 5.1 for $f''(w)$, I think the answer ...
2
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1answer
72 views

Inverse Fourier transform of Lorentzians and sign function

I'm trying to caculate the inverse Fourier transform of $$ G(\omega) = \dfrac{(\omega+a)^2+b^2}{((\omega-c)^2+b^2)((\omega+c)^2+b^2)} \mathrm{sgn}(\omega-d)$$ It is the product of two Lorentzians and ...
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3answers
120 views

Sum of Squares of Binomial Coefficients Using Residue Theorem

I ran across this interesting question recently that I have an idea for, but am unable to complete. Basically, we use the residue formula to find $$ \sum\limits_{k=0}^n {n\choose k}^2$$ We define $f$ ...
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1answer
71 views

Integrate $\int_{0}^{\infty} \frac{dx}{1 + x^n}$ using the Residue Theorem

I'm trying to compute the following integral: \begin{align} \int_{0}^{\infty} \frac{dx}{1 + x^n}, \quad n \geq 2. \end{align} Consider the function $f(z) = \frac{1}{1 + z^n}$. Consider the ...
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2answers
38 views

Evaluate $\int_{|z|=4}\tan z dz$

Evaluate $\int_{|z|=4}\tan z\,\mathrm dz$ $\tan z=\frac{\sin z}{\cos z}$ there is a a simple pole at $z=\frac{\pi}{2}$ $\operatorname{Res}(f,\frac{\pi}{2})=\lim_{z\to \frac{\pi}{2}}(z-\frac{\pi}{2})(...
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4answers
62 views

Evaluate $\int_{|z|=3}\frac{dz}{z^3(z^{10}-2)}$

$$\int_{|z|=3}\frac{dz}{z^3(z^{10}-2)}$$ There are singularities at $z=0$ and $z^{10}=2\iff z=\sqrt[10]{2}e^{\frac{i\pi k}{5}} \text{ where } k=0,1,...9$ so we need to find $10$ residues? or have ...
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2answers
38 views

Residue of $\frac{1}{(z^2-1)^3}$ at the singularities.

Find the residue of $$\frac{1}{(z^2-1)^3}$$ at the singularities. To find which order of pole it is I tried to look to the order of zeros of $(z^2-1)^3$ deriving the function seems to big the hard ...
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1answer
20 views

Residue of $ctg(z)$

Find the residue of $$ctg(z)$$ $cos(z)=\sum_{n=0}^{\infty}(-1)^n\frac{z^{2n}}{2n!}$ $sin(z)=\sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{(2n+1)!}$ So $ctg(z)=\frac{cos(z)}{sin(z)}=\sum_{n=0}^{\infty}(-...
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0answers
31 views

For any positive integer $n \in \mathbb{N}, \text{Res} \Big ( \frac{1}{(1-e^{-z})^n} ;0 \Big ) = 1$.

I am trying to prove the claim that: For any positive integer $n \in \mathbb{N}$, \begin{align} \text{Res} \Big ( \frac{1}{(1-e^{-z})^n} ;0 \Big ) = 1. \end{align} Here's my working: Let $f(z) ...
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1answer
53 views

Computing a residue

To find the value of $\int_0^\pi \frac{\sin^2 \theta}{a + \cos \theta} \, d\theta$ with $a > 1$ using residue theory, I am trying to compute the residue at the simple pole $z=-a+\sqrt{a^2-1}$ ...
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1answer
38 views

Find inverse Fourier transform of Lorentzian using complex integration

I am trying to prove the inverse Fourier transform relation of a Lorentzian $$F(\omega) = \dfrac{2b}{(\omega-a)^2+b^2} = \dfrac{1}{b+i(\omega-a)}+\dfrac{1}{b-i(\omega-a)}$$ using the relation $$f(t)...
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1answer
59 views

Finding the residue

Finding the fourth derivative in order to get residue seems me very complicated, is there another way? $$Res\left( z=i,\frac { { e }^{ iz } }{ { \left( { z }^{ 2 }+1 \right) }^{ 5 } } \right) =\lim ...
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1answer
55 views

Complex integral with $\int_{+\partial D}\frac{\sin\left(\frac{1}{z}\right)\cos\left(\frac{1}{z-2}\right)}{z-5}\mathrm{dz}$

Hi guys in this integral $$\int_{+\partial D}\dfrac{\sin\left(\dfrac{1}{z}\right)\cos\left(\dfrac{1}{z-2}\right)}{z-5}\mathrm{dz}$$ where $D=\{z\in\mathbb{C}:|z|<3\}$, is $z=5$ a pole, and are $...
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1answer
36 views

Complex integral $\oint_{|z|=1} \frac{z^2}{\sin^3 z \cos z} dz$

Hello I am trying to evaluate: $$I=\oint_{|z|=1} \frac{z^2}{\sin^3 z \cos z} dz$$ My try was to compute the residue at infinity since $I=2\pi i Res(f,\infty)$ . Now since $$\sin^3 z= \frac{3}{4}\sin z ...
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1answer
39 views

Laurent series for $f(z) = \frac{\sinh(z + 3i)}{z(z + 3i)^3}$ at $-3i$ (not leaving as a product of series)

I got this problem in my complex analysis class: Find the Laurent series of $$f(z) = \frac{\sinh(z + 3i)}{z(z + 3i)^3}$$ to calculate the residue at $z=-3i$. Is there an easy way to calculate the ...
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0answers
41 views

square roots in multiplicative group of integers modulo n

How do I exactly determine all solutions for square roots if they exists and what to do if for one square root exist two solutions? Let's for example take the Group $(\mathbb{Z}/7\mathbb{Z})$ with ...
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1answer
32 views

Calculating Residue of $f^2$ with pole of order 2

The question: The function $f$ has a Pol of order 2 in $z_0$. Calculate the residue of $f^2$ in $z_0$ using the Laurentcoefficients of $f$. My attempt: I tried to use the fact that if $f(z) = (z- ...
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2answers
30 views

Improper integrals with residue theorem [closed]

How to prove that when solving improper integrals with residue theorem, we should only include residues in the upper complex plane? I've tried working my way through with Jordan's lemma, but I got ...
2
votes
1answer
58 views

Residue identity for function composition: $ \operatorname{Res}(f; h(a)) = \operatorname{Res}((f∘h)h'; a)$

With $a \in \mathbb{C}$, let $h: D_1 \to D_2$ and $f: D_2\setminus{\{h(a)\}} \to \mathbb{C}$ be analytic functions. Also require $h'(a) \neq 0$. Claim: $$ \operatorname{Res}(f; h(a)) = \...
0
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0answers
34 views

Contour Integration of a simple expression

Let's calculate the integral $$ \oint_\gamma \frac{e^{ikz}}{z} \, {\rm d} z $$ where $k>0$ and $\gamma$ is the closed contour starting at $-\infty$ to $-\epsilon$, then the $\epsilon$ half-circle ...
5
votes
2answers
55 views

limit at infinity of a meromorphic function using residues

Let $z_1,\dots,z_n\in\mathbb{C}$ be different points and let $f:\mathbb{C}\setminus\{z_1,\dots,z_n\}\to\mathbb{C}$ be a holomorphic function. Suppose that $\lim\limits_{z\to\infty}f(z)=0$. Prove that ...
2
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0answers
73 views

Problem with computing residues

The problem is from Riley, Hobson and Bence's Mathematical Methods for Physics and Engineering. I followed the standard procedure, changing $x$ to $z$ and factorizing the denominator. Then the ...
0
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1answer
48 views

The Maclaurin Series for holomorphic function f

$\textbf{Exercise} \quad $Let $R>1$ and let $f$ be holomorphic on $\vert z \vert <R$ except at $z=1$, where $f$ has a simple pole. If \begin{align*} f(z)=\sum_{n=0}^{\infty}a_nz^n \quad (\...
0
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1answer
29 views

Complex Integral with Similar Integrands Yielding Very Different Results

According to Wolfram, the integral $\int_{-\pi}^{\pi} \frac{e^{1+iy}}{e^{1+iy}-1}dy$ equals $2\pi$, but $\int_{-\pi}^{\pi} \frac{e^{-1+iy}}{e^{-1+iy}-1}dy$ equals $0$. Can someone explain to me why ...
5
votes
4answers
172 views

Evaluating the complex integral $\int_{0}^{\infty} \frac {\sin (\ln x) dx }{x^2 + 4} $

I want to evaluate following integral: $\int_{0}^{\infty} \frac {\sin (\ln x) dx }{x^2 + 4} $ Obviously $x$ $ \gt $ $0$ and the function we want to integrate isn't even nor odd. And I need to ...