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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16
votes
2answers
452 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
118 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
118 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
175 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
125 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
184 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
54 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 ...
9
votes
3answers
304 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
36 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
181 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
150 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
151 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
68 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
71 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
114 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
81 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
56 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
46 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
110 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
181 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
82 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
148 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
336 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
141 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
107 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
81 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
108 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
173 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} = ...
12
votes
3answers
530 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
79 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
89 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
166 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
555 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
70 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
85 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
118 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
106 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
101 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
153 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
188 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 ...
5
votes
3answers
157 views

Calculating $\int_{0}^{\infty} x^{a-1} \cos(x) \ \mathrm dx = \Gamma(a) \cos (\pi a/2)$

My goal is to calculate the integral $\int_{0}^{\infty} x^{a-1} \cos(x) dx = \Gamma(a) \cos (\pi a/2)$, where $0<a<1$, and my textbook provides the hint: integrate $z^{a-1} e^{iz}$ around the ...
0
votes
1answer
39 views

Residue Theorem for Denominator with $e^z$

$$ f(z)=\frac{z^3}{e^z-1} $$ Is this a simple pole at $z=0$ or some other types of pole? If it is a simple pole, what is its residue? Is it using this formula or other else? $$ \lim_{z\to 0}=zf(z) ...
1
vote
1answer
56 views

Prove on residue theorem

I have try to use the equation $$ Res(f;z_0)=\lim_{z\to z_0}\frac1{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)] $$ But very soon I stuck, is that a good way to solve it?
2
votes
2answers
335 views

What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
1
vote
3answers
142 views

$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx$ using Complex methods

$$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx. $$ I am not sure how to solve this question. Can anyone help me to approach this problem. Thanks.
1
vote
2answers
173 views

Use the Residue theorem and its application to compute the integral

$$\int_{-\infty}^{\infty} \frac{x^2}{x^4-4x^2+5} dx. $$ I am not sure how to approach this question. Can anyone use the complex variable theory to help me solving the problem please? Thank you very ...
1
vote
1answer
191 views

Evaluating series by contour integration, the residue theorem, and cotangent

I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is ...