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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15
votes
0answers
401 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
0
votes
2answers
72 views

$\int_{0}^{\infty}\frac{\cos2\pi x}{x^4+x^2+1}dx=-\frac{\pi}{2\sqrt{3}}\mathrm{e}^{-\pi\sqrt{3}}$

Can somebody help me out with the following integral? Prove that: $\int_{0}^{\infty}\frac{cos2\pi x}{x^4+x^2+1}dx=\frac{-\pi}{2\sqrt{3}}e^{-\pi\sqrt{3}}$ I have already determined the ...
0
votes
4answers
58 views

Integrating $t^{2r-1} / t^{2k} (1+t^2)^{r+1}$

Let $k$ and $r$ be natural numbers such that $1 \leq k \leq r$. I want to calculate $$ \int_0^\infty \frac{t^{2r-1}}{t^{2k}(1+t^2)^{r+1}} dt. $$ Since the integrand is an odd function the standard ...
0
votes
0answers
96 views

Pole on path of integration.

Upon evaluating $$\int_0^\infty \frac{1}{1+z^5}dz$$ using the Residue Theorem, why isn't the pole at $z=e^{\pi i}$ taken under consideration in the summation of the residues? We were taught that ...
4
votes
4answers
320 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
0
votes
1answer
52 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
1
vote
2answers
161 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
16
votes
2answers
439 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
1
vote
1answer
104 views

Residue of a simple pole. Why are they different?

We'll show you two way of calculation of the Residue in consideration. $$f(z) = \frac{z\sin(z)}{1-\cos(z)}$$ I'm interested to calculate the residues in $2\pi$ and $-2\pi$. I choose one of ...
0
votes
1answer
109 views

Find the order of the poles of $\dfrac{z}{\cos z}$

I know that they are simple poles, but how can you find this? The usual equation that I have for finding the order of poles which is, $$ \displaystyle\lim_{z\to z_0} (z-z_0)^{n}f(z), $$ and seeing ...
3
votes
1answer
152 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
1
vote
3answers
118 views

Question Residues -integral at Complex Analysis

How can i find the integral below , which transformation should i do ? İ think i need to get $sin$ and $cos $ but i can't see $$\int\limits^{+\infty}_{-\infty} \frac{ \exp\left({ax}\right)} ...
3
votes
1answer
165 views

Choice of branches for contour integration.

Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of $f(z)$ are $z\leq 0$ (presumably for the logarithmic term), and ...
1
vote
2answers
50 views

Residue of a a complex quotient

I have the following Laurent expansion corresponding to the function: $$f(z)=\frac{z+2}{z^2-4iz-3}$$ $$f(z)=\left(-1+\frac{1}{2}i\right)\sum_{n=1}^\infty ...
8
votes
3answers
291 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
1
vote
1answer
35 views

Carry out integral by using Cauchy's theorem

I have kind of a silly question, which probably has an easy answer which I should know myself, but here goes. Say we want to integrate $$ \int_{-\infty}^\infty dx \frac{1}{(x^2 + 1)(x - 1 - i)}. $$ If ...
3
votes
2answers
168 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
3
votes
0answers
122 views

How to calculate this residue

How to calculate this residue $$Res\left(\frac{\ln z}{z(z+1)},0\right).$$ Is it $\infty$? And if this could not be calculated, then how to calculate $$\int_0^\infty \frac{x}{e^x+1}dx$$ by changing ...
4
votes
2answers
123 views

Intuition behind the residue at infinity [duplicate]

The residue at infinity is given by: $$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$ Where $f$ is an analytic function except at finite number of singular points ...
2
votes
1answer
61 views

Integral using residue theorem

We have the following problem given: $$ \int_{-\infty}^\infty \frac{\cos(t)^2}{t^4 + 5 t^2 + 4} \, \mathrm dt. $$ I thought that I could solve it using the residue theorem and by arguing that the ...
1
vote
1answer
67 views

Limit of characteristic function

I have a characteristic function defined as the following: $$\phi(\frac{t}{N})^N = (\alpha E_{\alpha+1}(-\frac{iLt}{N}))^N$$ where $E_n(z)$ is the $E_n$ function having the form $E_n(z) = ...
6
votes
3answers
113 views

calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
1
vote
2answers
79 views

question on integrals

Let $\displaystyle A=\int_0^1 \frac{dx}{1+x^8}$. Then which of the following are true: 1) $A\lt 1$, 2) $A\gt 1$, ...
2
votes
2answers
55 views

Problem with Mellin Barnes type integral

Using the Mellin Barnes technique for a certain Feynman integral, I arrive at $$ I= \frac1{2\pi i} \int\limits_{-i\infty}^{i\infty} dz\; \Gamma^4\left(\frac12 + z\right) ...
0
votes
1answer
43 views

Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
2
votes
1answer
96 views

Residue Theorem for trigonometric integrals.

I am working on the following statement. Let $Q = Q(x,y): \mathbb R^2 \to \mathbb R$ be a rational function, which is continuous on the unit circle $S_1(0)$. Let furthermore $f: \mathbb C \to ...
2
votes
2answers
162 views

How to show the residue of an analytic function's derivative is equal to zero?

Let $r>0$ . for $f: \Bbb D_r(0)-{0}\mapsto \Bbb C$ analytic function show that $Res(f';0)=0$ we know by residue therom $∫_Cf'(z)dz=2iπRes(f',0)$ What property of analytic functions will we use? ...
2
votes
1answer
78 views

Computing the residues for $1 /( z^2\sin(z))$

I am trying to find the residues for the function $1 / (z^2\sin (z))$. By expanding the function around the singularites I managed to find $\text{Res}(f;0) = 1/3!$. There is also a singularity at $z = ...
7
votes
1answer
146 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= ...
7
votes
1answer
249 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
1
vote
1answer
139 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
1
vote
2answers
94 views

how to find residues of $\frac{e^{st}}{\cosh(a\sqrt{s})}$?

Can someone give me a hint on how to find residues of $\frac{e^{st}}{\cosh(a\sqrt{s})}$ ? I am trying to solve an integral using residue method. (actually inverse Laplace transform). $a$ is real in ...
2
votes
2answers
77 views

Residue theory complex

$$\int_{-\infty}^{\infty}\frac{\cos x}{x^4+5x^2+4}dx$$ Give full justification of your answer, including appropriate bounds for the contributions from all portions of your contour! I am not ...
1
vote
1answer
37 views

Discrepancy in counting the number of poles in complex function when refactoring

If I have a function that looks like this: $$f(z) = \frac{(z-i)^2}{\sin^2z}$$ and I want to find its poles within the unit circle contour, $|z| = 1$, it seems from this equation that there is a pole ...
4
votes
1answer
104 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
1
vote
1answer
154 views

Evaluating real improper integral by residues

I've been trying to solve this integral and have been getting nowhere: $$ \int_0^\infty \frac{dx}{(1+x^2)x^a} \;,\; 0<a<1 $$ The solution says that $$ \int_0^\infty \frac{dx}{(1+x^2)x^a} = ...
11
votes
3answers
468 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
1
vote
0answers
65 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
2
votes
2answers
88 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
1
vote
5answers
130 views

Indented Path Integration

The goal is to show that $$\int_0^\infty \frac{x^{1/3}\log(x)}{x^2 + 1}dx = \frac{\pi^2}{6}$$ and that $$\int_0^\infty \frac{x^{1/3}}{x^2 + 1}dx = \frac{\pi}{\sqrt{3}}.$$ So, we start with the ...
4
votes
2answers
395 views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
1
vote
1answer
68 views

Contour integration in the complex plane gone wrong

Considering a function of complex variable $z$: $$f(z)=\frac{e^z}{z}$$ and a contour integral: $$\oint_C dz f(z)$$ such that the contour $C$ encircles the origin counterclockwise, it is clear from the ...
4
votes
2answers
83 views

Establish $\int_0^{\infty} \frac{x^a}{x^2 + b^2}dx = \frac{\pi b^{a-1}}{2 \cos(\pi a /2)}$ when $-1 < a < 1$

My attempt at a solution: (this is homework, btw) Let $f(z) = \frac{z^a}{z^2 + b^2}dz$ then the singularities of $f$ occur at $\pm ib$. $$ Res(f; ib) = \frac{z^a}{z + ib} \biggr |_{ib} = ...
0
votes
0answers
39 views

perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$ f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: is it possible to compute ...
3
votes
5answers
107 views

Residues at singularities

I have the following question: Show that the integral $$\int_{-\infty}^{+\infty}\frac{\cos\pi x}{2x-1}dx = -\frac\pi2$$ Clearly there is a singularity at $z=1/2$ but I think this is a removable ...
0
votes
2answers
96 views

Find the poles and residues

Find the poles and residues of $\frac{z \ln(z)}{(z^2+1)(z-c)}$, where $c$ is a real positive constant. I've found the poles to be $z=i$, $-i$ and $c$. These are simple poles. How do I now ...
3
votes
3answers
92 views

Calculate this residue

I'm kind of strigling with a problem right now. It is as follows: Calculate the residues of this function at all isolated singularities. $$f(z)=\frac{e^z}{\sin^2z}$$ I got the singularities ...
0
votes
1answer
48 views

Verity of Residue theorem of [0,2pi]

After I turn $$ cos\theta=\frac12(z+\frac1{z})$$and $$ d\theta=\frac1{iz}dz$$ the denominator become a mess $$ \frac{dz}{(a^2+\frac{b^2}4(z^2+2+\frac1{z^2})+\frac{ab}2(z+\frac1z))(iz)}$$ How can a ...
2
votes
2answers
148 views

Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
1
vote
3answers
151 views

Find the residue of $\frac{e^{iz}}{(z^2+1)^5}$ at $z = i$ and evaluate $\int_0^{\infty} \cos x/(x^2+1)^5 dx$

I know the evaluation of $\int_0^{\infty} \cos x/(x^2+1)^5 dx$ requires that I solve the first part, but for some reason I'm stumped. I get that I should use $\lim_{z \to ...