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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Residue theorem pole

How do I find poles of this function? $$f(z) = \frac{1-\cos z}{z^4 +z^3}$$ I am unable to identify zero pole order of $f(z)$ as it includes trigonometric functions and I'm new to this concept. Please ...
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28 views

Bounding a polynomial function

Suppose $P(z)$ and $Q(z)$ are polynomials satisfying $deg(Q) \geq deg(P) +2$ and $\gamma_R$ is an arc parameterized by $z=Re^{i\theta}$ for $\theta \in [\alpha,\beta]$. Show that $ \lim_{R \to \infty}\...
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48 views

Solve $\int_{-\infty}^{\infty} \frac{ e^{ -\frac{1}{2} \frac{\rho}{\beta} (t - j \alpha \frac{\rho}{\beta})^{2} }}{(t-j \rho)^{M} (t+j \rho)^{M}} dt$

I want to solve the integral: $$ f(x) = \int_{-\infty}^{\infty} \frac{ e^{ -\frac{1}{2} \frac{\rho}{\beta} (t - j \alpha \frac{\rho}{\beta})^{2} }}{(t-j \rho)^{M} (t+j \rho)^{M}} dt $$ where $j$ is ...
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2answers
43 views

Contour integration around pole and essential singularity

I've got the following integral: $$ \frac{1}{2\pi i}\oint_\limits{|z|\,=\,R} \frac{z^5}{z-1} e^\frac{2}{z} dz, $$ where $R$ - sufficiently large number. I've tried to evaluate it with residues. The ...
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1answer
30 views

Trigonometric integral using residue complex analysis.

There is this problem where I need to proof the following statement: $$\int_{0}^{\pi}\cos^{2n}\theta d\theta = \pi\frac{(2n)!}{2^{2n}(n!)^2}$$ First I used the fact that the function is ever so I can ...
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1answer
32 views

Residue theorem of rational function with real roots in the denominator

I am studying residue calculations and until now had learned that one should avoid using it for integral of rational functions P/Q when Q has real roots. However, I found an exercise concerning this ...
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3answers
61 views

How to prove $\int_{0}^{\infty} \frac{\sin x}{x(x^2+1)}dx=\frac{\pi}{2}(1-\frac{1}{e})$ by residue calculus?

How to prove $$I=\int_{0}^{\infty} \frac{\sin x}{x(x^2+1)}dx=\frac{\pi}{2}(1-\frac{1}{e})\, ?$$ Let $$f(z)=\frac{\sin z}{z(z^2+1)}=\frac{e^{iz}-e^{-iz}}{2i\,z(z^2+1)}.$$ Then I have no idea how to ...
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1answer
30 views

Solution verification of a question in Complex analysis (related to residues)

I am trying questions of complex analysis of an institute in which I don't study and I am confused in options so I thought I should post here. Let X be closed and continuously differentiable path in ...
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1answer
59 views

How to show orthogonality of the Laguerre polynomial $P_n(x)$?

At school, they ask me to solve this question: For $n \in \mathbb{N}$ and $x > 0$ we define $P_n(x) = \frac{1}{2\pi i}\int_{\Sigma}\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^t dt$ where $\Sigma$ is a ...
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2answers
53 views

Complex analysis for real valued integrals.

I'm asked to compute $$\int_{-\infty}^{\infty}\frac{\sin(ax)}{x(1+x^4)}dx$$ To do this, I consider the function $$f(z)=\frac{e^{aiz}}{z(1+z^4)}$$ and the contour composed by the segment on the real ...
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1answer
53 views

Contour integral in presence of a branch cut.

I'm trying to evaluate $$\int_0^{\infty} \frac{\log x}{1+x^3}dx$$ and I have to use Residue Theorem. I chose the classical pacman contour centered in the origin, and by using little/big circle ...
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3answers
83 views

Evaluating real integral using complex analysis.

I'm trying to compute the following integral: $$\int_0^{\infty}\frac{\sqrt{x}}{1+x^4}dx$$ I'll not write down everything I've done, but choosing the branch cut on the positive real axes we have that: $...
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1answer
34 views

integral using residue theorem $\int_{-\infty}^{\infty}\frac{\cos(x)dx}{(x^2 +a^2)(x^2 + b^2)}$ [closed]

i just need to solve this integral using residue theorem. $$\int_{-\infty}^{\infty}\frac{\cos(x)dx}{(x^2 +a^2)(x^2 + b^2)}$$ where $-1<a<1$. I just know i have to separate $\frac{\cos(x)dx}{(x^2 ...
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19 views

If $|f(z)|<k|z|^{-c}$, then $f(z)=\int_{-\infty}^{+\infty}\frac{f(t)}{t-z}dt$

The problem is: If $f$ is analytic in $\{z\in\mathbb{C}:\text{Im}(z)\geq 0\}$ and there exists $k,c>0$ such that $|f(z)|<k|z|^{-c}$ for all $z\neq 0$, then $f$ has the form $$f(z)=\dfrac{1}{2\pi ...
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1answer
38 views

How to get ILT of $e^{-\alpha \sqrt{s}}/s^{3/2}$ by residue theorem?

I posted some examples before such as $$\mathscr{L}^{-1}\left[\frac{e^{-\alpha \sqrt{s}}}{\sqrt{s}(s-A)}\right]=\frac{e^{-\alpha \sqrt{A}+At}}{\sqrt{A}} - \frac{1}{\pi}\int_{0}^{\infty}\frac{\cos{(\...
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1answer
38 views

Finding the residue at $z=z_0$ of $f(z)$ given…

i need to resolve this problem: Given $f(z)=\frac{1}{\big(g(z)\big)^2}$, show that if $g(z_0)=0$ and $g'(z_0)\neq 0$ then $f$ has a second order pole at $z_0$ and $\text{Res}_{z=z_0}f(z)=-\frac{g''(...
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1answer
35 views

Residue Calculus with different enclosing methods of contour

Consider the following integral: $$\frac{1}{2\pi i }\oint_C f(z) dz$$ with $$f(z)= \frac{z^2+1}{z^3}$$ and $C$ is the unit circle centered at the origin. Consider two ways to do it: Enclosing the ...
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78 views

complex integral involving branch cuts

[contour for given integral ][1] We have to integrate $\int_{0}^{a} \frac{log(x)\sqrt{a^2 -x^2}}{x^2 -b^2} dx$; $a>b>0$ .Since given function has branch points at $\pm a,0,\infty$ and poles at$\...
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24 views

Complex analysis (residue theorem)

With $g(z)=\begin{cases} \frac{\beta}{\exp (\beta z)+1} \\ \frac{\beta}{\exp (\beta z)-1} \end{cases}$ How do I compute $\operatorname{Res } \left(\ln(z-\epsilon)g(z)\right)$? Because of the log term ...
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2answers
53 views

Find the residue of $1/(z^2+1)^{n+1}$ at $z=i$.

I'm trying to find the residue of $\frac{1}{(z^2+1)^{n+1}}$ at $z=i$. I believe this should be approached combinatorially but I am getting stuck in the combinatorics. Here's what I have so far. $$ \...
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1answer
40 views

Computing real definite integral using residue theorem

I am trying to solve integral $\int_{-∞}^\infty \frac{cos(x)dx}{(x^2+a^2)(x^2+b^2)}$ using residue theorem. I found example of that with very similair integral, which is $\int_{0}^\infty \frac{cos(x)...
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23 views

Using summation theorem to evaluate alternating sum

So, I have been given the Summation Theorem and asked what should I replace $cot(\pi z) with $ if I want to evaluate $\sum_{n} (-1)^{n} f(n)$. This is from a course in complex analysis. Can someone ...
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1answer
43 views

Residue of $\frac{1}{(1+\cos z)^2}$ for $z_0=\pi$

I'm trying to find the residue of $f(z)=\frac{1}{(1+\cos z)^2}$ around $z_0=\pi$. I know that $z=\pi$ is a pole of order 4, so I'm using the formula: $$\frac{1}{(4-1)!}\lim_{z\to\pi}\frac{d^3}{dz^3}\...
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3answers
57 views

Contour integration to evaluate a real-valued integral

I am evaluating this integral: $$\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2+1)^2}\,dx$$ with the formula $$\int_{-\infty}^{\infty} f(x) \sin(sx) dx = 2\pi \sum\text{Re } \text{Res}[f(z) e^{isz}]$$ ...
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33 views

How to compute $\int_{0}^\infty \frac{x^a}{1+2x\cos(b)+x^2}dx$ by residue calculus?

How to compute $$\int_{0}^\infty \frac{x^a}{1+2x\cos(b)+x^2}dx$$ with $-1<a<1$, $a\neq0$, $-\pi < b < \pi$, and $b\neq 0$. First, I set $$f(z) = \frac{z^a}{1+2z\cos(b)+z^2}$$ Then how do I ...
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3answers
66 views

How to compute $\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx$?

I want to compute $$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx$$ My approach is $$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx=\int_{-\infty}^{\infty} \frac{1}{(x -\pi)^2}dx+\...
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3answers
68 views

Compute $\int_{-\infty}^{\infty} \frac{1}{(x-5)^2}dx$

I want to compute $$I = \int_{-\infty}^{\infty} \frac{1}{(x-5)^2}dx.$$ We can use the following methods Compute directly $$I = -(x-5)^{-1}\bigg |_{-\infty}^{\infty}=-\bigg(\frac{1}{x-5}\bigg)_{-\...
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1answer
63 views

Laplace Inversion with Residue Theorem doesn't satisfy IC of IVP

I have the following initial value problem $$ \frac{d\theta}{dt} = A(p-\theta) + B(\omega-\theta) $$ subject to the initial condition $$ \theta(0)=\theta_0 $$ and the constitutive set of equations $$ \...
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1answer
48 views

What is value of following integral ( Solve by complex analysis)

I am trying exercises of textbook Ponnusamy and Silvermann and I am struck on this problemof Section 9.37. Evaluate the following integral along different simple closed curves not passing through 0, +...
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1answer
20 views

How to determine the isolated singularity?

problem: Calculate integral : $\int _0 ^\infty \frac{\log x}{x^a(x-1)}dx$ 0<a<1 We should determine the isolated singularity,i think it only possibly has 0.since we know $\lim_{z\ to 1} \frac{\...
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0answers
36 views

Integral of real exponential and simple pole using residue theorem

Evaluate the integral, $$\int_a^\infty dx \frac{1}{x-b-i0}\exp\big(-\sqrt{x^2-a^2}\big),$$ where $0<a<b<\infty$, and $i0$ is a positive infinitesimal imaginary number, shifting the pole ...
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2answers
57 views

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (residue theorem)

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (although we may use normal real integral to solve, I wonder if contour analysis can also help?) The question offers a hint ...
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1answer
30 views

Finding the residue of a mod of a function…

What is the residue of $$f(z)=\frac{1}{|z+c||z-c|}$$ at $z=c$ and $z=-c$. I know to find the residue without mod in the denominator, but I have no idea of finding the residue with a mod in the ...
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0answers
21 views

Integral Value through Complex Integration (residue theorem)

I'd like to know how to evaluate the integral $$I=\int_0^\infty\frac{e^{-s^2}\sin(s)}{s}\,ds=\frac{\pi}{2}\text{erf}(1/2)$$ through the residue theorem. My first steps were to expand $\sin$ as ...
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1answer
47 views

By application of calculus of residues, can you please solve this problem?

By application of calculus of residues, prove that $$ \int_{0}^{2\pi} \frac{\cos^{3}\left(3\theta\right)} {1 - 2p\cos\left(2\theta\right) + p^{2}} \,\mathrm{d}\theta = \frac{\pi\left(1 - p + p^{2}\...
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1answer
31 views

ILT of hyperbolic functions

I want to calculate the ILT of this function with the residue theorem: $$F(s) = \frac{\cosh{(\alpha \sqrt{s})} \operatorname{csch}(\beta \sqrt{s})}{\sqrt{s}(s-\theta)}.$$ The branch cut is along the ...
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1answer
54 views

Solving the Integral: $f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt$ when $L$ is odd

I want to solve the following integral when $L$ is odd: $$ f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt $$ which can be simplified to: $$ f(...
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19 views

How to realize ILT by residue theorem?

$$F(s)=\frac{e^{-x \sqrt{s}/\sqrt{D}} (e^{2C \sqrt{s}/\sqrt{D}} + e^{2x \sqrt{s}/\sqrt{D}})}{s^{3/2} (e^{2c \sqrt{s}/\sqrt{D}} -1)}$$ By the residue theorem. I tried, but didn't realize. The ...
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0answers
37 views

Using Residue to solve partial fraction decompositions

a) $Res(3i)$ for $R(z) = \frac{z^2-9}{(z^2+9)^2}$ So I solved this by using the formula is: $\lim_{z \to 3i} \frac{1}{1!} \frac{d}{dz} [(z^2+9)^2 \times \frac{z^2-9}{(z^2+9)^2}]$ $\lim_{z \to 3i} \...
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1answer
67 views

How to do this partial fraction decomposition (Complex numbers)

The question states: Let $R = \frac{P}{Q}$ be a rational function with $\deg P < \deg Q.$ If $ζ$ is a pole of $R$, then the coefficient of $1/(Z-ζ)$ in the partial fraction decomposition of $R$ is ...
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34 views

Non-isolated singularity and contour integral

I encounter contour integrals of the following form, $$ \oint_{|z| = 1} f_q(z) \frac{ dz }{ 2\pi i}\ , \qquad |q| < 1 $$ where the meromorphic function $f_q(z)$ contains a lot of simple poles of ...
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2answers
73 views

Evaluate the integral (sinx-x)/x^3 based on residue [duplicate]

The original quesion is $$\int_0^\infty\left(\frac{\sin x}{x^3}-\frac1{x^2}\right)\,dx$$ Can I divide them into two parts? Then using residue theorem?
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1answer
56 views

Computation of a certain contour integral

I have to do the following integral(using complex analysis): $$\int_{0}^{\infty} \frac{\cos{nx}}{x^{4}+1} dx $$ So, first I evaluated $x^{4}+1=0 $ and got $x = \pm \frac{1+i}{\sqrt{2}}, \pm \frac{-1+i}...
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0answers
42 views

Contour integral of fractional function

So, I have to solve the following integral $$ \int_{0}^{\infty} \frac{\sin^{2}{x}}{x^{2}} dx$$ I'm aware that this has been asked about before on this site, but I want comments on my attempted ...
2
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1answer
73 views

Question about complex integrals

I have two questions about integrals. 1.) $$\frac{1}{2\pi i} \int_{\gamma} \frac{dw}{\sin{\frac{1}{w}}}$$ where $\gamma$ is the circle $|w| = \frac{1}{5}$. 2.) $$\int_{\gamma_{a}} \frac{z^{2} + e^{z}}{...
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2answers
48 views

Polynomials arising from residue of $\frac{z^m}{(1+z^N)^k}$ at $e^{\frac{i\pi}N}$

I have been trying to find a general expression for $${I=\int_0^{\infty}}\frac{x^m}{(1+x^N)^k}dx$$ where $m,N,k\in\mathbb{N}_0$ , $m\leq{Nk-2}$ , $N\geq2$ and $k\geq1$. To do so I have been using ...
3
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0answers
29 views

Is there a discrete version of the residue theorem?

Consider the integral: $$\int\limits_{-\infty}^{\infty} \frac{1}{(x+iz)}\frac{1}{(y+iz)} dz = \frac{-2\pi}{|x-y|} $$ if $xy<0$ and zero otherwise. i.e. it is only non-zero if $x$ and $y$ are ...
2
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1answer
51 views

$\int_0^{\infty}e^{-t^2/2}\,\frac{e^{2\pi}-\cos\left(\sqrt{2\pi} t\right)}{e^{4\pi}-2e^{2\pi}\cos\left(\sqrt{2\pi} t\right)+1} dt $

How does one show $$ \int_0^{\infty} e^{-t^2/2} \left[ \frac{e^{2\pi} - \cos\left(\sqrt{2\pi} t \right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} t\right) + 1} \right] dt = \...
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1answer
61 views

Evaluate : $\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$

Computer $$\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$$ Where : $$\gamma =\{ |z|=1~ ; ~\Re z≥0,\Im z≥0 \}$$ I try : $z=e^{it}$ then $dz=ie^{it}dt$ And $t\in [0,\frac{π}{2}$ then ...
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2answers
47 views

Complex Integration (Residue Theorem)

How do I integrate $ \oint_{C:\left | z \right |= R}^{}\frac{e^{^{\frac{1}{z}}}}{z^{2}+1}dz $ , with $ 0< R< 1 $ ? I am supposed to use the residue theorem but there's no Laurent series around z=...

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