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.

Filter by
Sorted by
Tagged with
1
vote
2answers
43 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 ...
0
votes
1answer
23 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 ...
0
votes
0answers
19 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 ...
0
votes
1answer
45 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}\...
1
vote
1answer
26 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 ...
0
votes
1answer
49 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(...
0
votes
0answers
17 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 ...
1
vote
0answers
35 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} \...
-2
votes
1answer
60 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 ...
0
votes
0answers
31 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 ...
0
votes
2answers
68 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?
0
votes
1answer
54 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}...
1
vote
0answers
41 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
votes
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}}{...
1
vote
2answers
46 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
votes
0answers
28 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
votes
1answer
44 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 = \...
0
votes
1answer
59 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 ...
0
votes
2answers
44 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=...
0
votes
0answers
23 views

how to integrate this by using contour integral and residue

$\int_{x=-\infty}^{\infty} \frac{\beta\alpha^\beta(1-\alpha)}{\pi(1-\alpha^\beta)}e^{itx} \frac{[(0.5+\frac{1}{\pi}\arctan(x))(1-\alpha)+\alpha]^{-(\beta+1)}}{(1+x^{2})}~~-\infty\leq x\leq \infty,\...
1
vote
1answer
27 views

Calculating the inverse Laplace transform of $F_{5}(s)=\frac{-A_{10} e^{-A_{7} \sqrt s }}{(\sqrt s+A_8) \sqrt{S+\theta_c}(S-A_4)}$ by residue theorem

I have calculated the ILT by residue theorem, but the result is wrong. Would you like to check the calculation. Thanks.
1
vote
0answers
32 views

Residue of $\frac{e^{-ikz}}{\cosh z}$

I am asked to compute the residue of $$\frac{e^{-ikz}}{\cosh z}$$ Consider the zeros of $\cosh z$: $$z=\frac{\pi(2n+1)i}{2}$$ Now consider: $$\lim_{z\rightarrow\frac{\pi(2n+1)i}{2}}{\Bigr(z-\frac{\pi(...
3
votes
0answers
111 views

Complex integration $\int_{-\infty}^{\infty}\frac{e^{cx}}{1+e^x}\,dx$ for $c \in (0,1)$ [duplicate]

I have to calculate real integral $\displaystyle \int_{-\infty}^\infty \frac{e^{cx}}{1+e^x} \, \text{d}x$ for $c \in (0,1)$. I have also hint, to integrate over the squere, which consists of $x$ axis, ...
0
votes
0answers
22 views

Residues of $\frac{1}{z\sinh(z)^2}$

I can manage the triple pole at $z=0$ by finding the Laurent expansion, but the double poles at $k\pi i$ I'm not sure how to find.
1
vote
0answers
31 views

Evaluating contour integrals over the unit circle of rational functions.

Let $p(z):= a_0 + a_1z + \cdots + a_nz^n$ be a degree $n$ polynomial, let $m$ be a large integer (which we may assume much larger than $n$), and let $k$ be some integer in the range $m+1, \cdots, m+n$....
6
votes
1answer
70 views

Residue theorem for $ I=\int_{-\infty}^{+\infty}\frac{e^{\mathrm{i}\,t\,z}}{(z-z_1)(z-z_2)} \, \mathrm{d}z$

If I use the residue theorem to evaluate the integral $$ I(t)=\int_{-\infty}^{+\infty}\frac{e^{\mathrm{i}\,t\,z}}{(z-z_1)(z-z_2)} \, \mathrm{d}z$$ with $t>0$, $\mathrm{Im}(z_1)>0$ and $\mathrm{...
1
vote
1answer
55 views

Find the value of $~\int_0^{2\pi}\frac{d\theta}{1-2a\cos\theta + a^2}~~~$ for $~|a|<1~.$ [duplicate]

Using residue theorem, find the value of $$~\int_0^{2\pi}\dfrac{d\theta}{1-2a\cos\theta + a^2}$$ for $~|a|<1~.$ I know that the value of the integral is $~\frac{2\pi}{1-a^2}~$(I found it by using ...
5
votes
1answer
115 views

Detail on the the choice of sign when computing $\int_{-1}^1\sqrt{1-x^2} \, dx$ by residues

Say I want to compute $\int_{-1}^1\sqrt{1-x^2}\,dx$ by residues. I use the traditional dog bone contour. The small circuits around $-1$ and $1$ do not contribute, while the segments in the middle add ...
0
votes
1answer
55 views

How to find the residue of an even function at a pole of order $~n~?$

Question: How to find the residue of an even function at a pole of order $~n~?$ I know how to find the residue of a function depending on the nature of the singularity. But the above question stuck ...
1
vote
0answers
11 views

What is the generalisation of this residue theorem to N-dimensions?

Consider the contour integral equation: $$\int_C \frac{1}{(z-w)} \frac{1}{(w-v)} dw = \frac{\pi}{z-v}$$ Where $C$ is a contour that completely surrounds the complex point $z$ or $v$. Is there a ...
0
votes
2answers
50 views

How to calculate $\int_C \cos \big( \cos \frac 1z \big) dz$?

The question says: If $C$ is a closed curve enclosing origin in the positive sense. Then $\int_C \cos \big( \cos \frac 1z \big) dz=$ ? $(1)\quad 0$ $(2)\quad 2\pi i$ $(3) \quad \pi i$ $(4)\quad -\pi i$...
1
vote
0answers
47 views

how to calculate $\int_0^{\infty}\frac{\sin^3(x)}{x^3}$dx [duplicate]

I need help for how to evaluate the following integral $\int_0^{\infty}\frac{\sin^3(x)}{x^3}dx$. Using wolframalpha, its value is $\frac{3\pi}{8}$. So I guess that I can use residue to solve this ...
3
votes
1answer
95 views

Improper Definite integral $\int_{-\infty}^\infty -\frac{i \pi e^{-i a p} \text{sech}\left(\frac{c p}{2}\right)}{p}dp$

I came across this improper integral that I couldn't solve $$\int_{-\infty}^\infty -\frac{i \pi e^{-i a p} \text{sech}\left(\frac{c p}{2}\right)}{p} dp$$ My guess would be to use Residue theorem but ...
3
votes
1answer
88 views

Calculate $\int_{0}^{\infty} \frac{x-\sin(x)}{x^3(1+x^2)}$

Calculate $$\int_{0}^{\infty} \frac{x-\sin(x)}{x^3(1+x^2)}$$ I know i am supposed to use residue theorem. However, I am having trouble with the pole at $z=0$ normally i would try the funciton $$f(z)=\...
0
votes
2answers
26 views

Residue calculation check.

The $\require{cancel}\operatorname{Res}\left( \frac{z^a}{(z^2 + 1)^2},i \right)$ = $\frac{(1-a)e^{a i \pi/2}}{4i}$ is supposed to be this where $a \neq 1$ and $-1 < a < 3$ But what I am getting ...
0
votes
2answers
86 views

Mistake in calculating $\int_{-\infty}^{\infty}\frac{\sin(x)}{x(1+x^2)}$

Mistake in calculating $\int_{-\infty}^{\infty}\frac{\sin(x)}{x(1+x^2)}$ I want to use Residue theorem. Consider the function $$f(z)=\frac{e^{iz}}{z(1+z^2)}$$ We integrate it over a semicircle $C_R$ ...
2
votes
0answers
53 views

An Integral of a Bessel function

I have an integral of $\ R_{1} =-i\int_{t}^{\infty }\frac{ds}{s}J_{1}\left( 2s\right) e^{i\nu s}$, where $J_{1}\left( 2s\right)$ is the Bessel function of the first kind. and $t$ is a positive number. ...
5
votes
3answers
294 views

calculate: $\int_{-\infty}^{\infty}\frac{\cos\frac{\pi}{2}x}{1-x^{2}}dx$ using complex analysis ; detect my mistake

calculate: $\int_{-\infty}^{\infty}\frac{\cos\frac{\pi}{2}x}{1-x^{2}}dx$ using complex analysis. My try: $\int_{-\infty}^{\infty}\frac{\cos\frac{\pi}{2}x}{1-x^{2}}dx$ symetric therefore : $ \int_{-\...
0
votes
0answers
16 views

Determining a meromorphic function by its poles.

Given a series of complex numbers $\{a_j\}_{j\in\mathbb{Z}}$, there exists a meromorphic function having simple poles only at all $\mathbb{Z}$ with the residues equal to $\{a_j\}_{j\in\mathbb{Z}}$ ...
1
vote
4answers
187 views

Calculate:$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$ with contour integration

Calculate: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$ My try: Keyhole integration: $\displaystyle \frac{\pi i\ln R\cdot e}{(Re^{\theta i}+1)^{3}}\rightarrow 0$ (we take $r$ as large as we ...
0
votes
0answers
26 views

Inverse Fourier transform in two dimensions

I'm trying to evaluate the following inverse Fourier transform: $$\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\bar{k}, \bar{l}) e^{\iota( \bar{k} \bar{X}+\bar{l} \bar{Y})} d \...
1
vote
2answers
86 views

calculate $\oint_{|z|=1} \left(\frac{z}{z-a}\right)^n \, dz$

calculate $\oint_{|z|=1}\left(\frac{z}{z-a}\right)^{n}$ whereas a is different from 1, and n is integer. My try: \begin{align} & \oint_{|z|=1}\left(\frac{z}{z-a}\right)^n \, dz\\[6pt] & \oint_{...
3
votes
2answers
171 views

Calculate: $\int_{0}^{\infty}\frac{\sin x}{x^{3}+x}\mathrm{d}x$ ;find my mistake

Calculate: $$\int_{0}^{\infty}\frac{\sin x}{x^{3}+x}\mathrm{d}x$$ My try: Let's split to a pacman style path, with little circle around the singularity and 2 rays from $0$ to $\infty$: $\displaystyle \...
0
votes
1answer
71 views

Complex residue theorem integral

How would I apply the residue theorem to work out the following integral? \begin{equation} \int_{0}^{2\pi} \frac{\sin^2\theta\, d\theta}{5-4\cos\theta} \end{equation} I've looked into the residue ...
1
vote
1answer
48 views

Sign of the line integral ($\int_{\vert z\vert=1} {1 \over z^2} \tan({\pi \over z}) dz$)

Find the value of the $$\int_{\vert z\vert=1} {1 \over z^2} \tan\left({\pi \over z}\right) dz$$ When we substitute $\omega = {1 \over z}$, then $d\omega = - {1 \over z^2}dz$, hence $\int_{\vert z\...
4
votes
2answers
161 views

calculate: $\int_0^\infty \frac{\log x \, dx}{(x+a)(x+b)}$ using contour integration

given $ a\neq b;b,a,b>0 $ calculate: $\int_0^\infty\frac{\log x \, dx}{(x+a)(x+b)}$ my try: I take on the rectangle: $[-\varepsilon,\infty]\times[-\varepsilon,\varepsilon]$ I have only two simple ...
1
vote
3answers
65 views

Showing Existence of Antiderivative for Complex-Valued Function

I am asked to show that for $z\in \mathbb{C} \setminus \{0,1\}$, there exists an analytic (single-valued) function, $F(z)$ on $\mathbb{C} \setminus \{0,1\}$, such that $F'=f$, where $$f(z) = \frac{(1-...
-1
votes
2answers
109 views

calculate: $\int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta$

calculate: $\int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta$ my try: $ \begin{array}{c} \int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta\\ \int_{0}^{2\pi}e^{\cos\theta}(\...
1
vote
1answer
68 views

calculate $\oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}$

calculate $\oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}$ my try: $ \begin{array}{c} \oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}\\ \oint_{|z|=1}z^{2018}{\displaystyle \...
2
votes
3answers
106 views

why this claim is wrong ? (as it leads to $\int_{0}^{\infty}\frac{\sin x}{x}dx=0$)

let's define $\begin{array}{c} f=\begin{cases} \frac{\sin x}{x} & x\neq0\\ 1 & x=0 \end{cases}\end{array}$ f is holomorphic on $\mathbb{C}$ as it equals to it's taylor series. therefore, for ...

1
2 3 4 5
43