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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10
votes
2answers
377 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
62 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
86 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
116 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
308 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
61 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
81 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
36 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
102 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
89 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
81 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
43 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
143 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
131 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
132 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
35 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
50 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
219 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
134 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
130 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
137 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 ...
5
votes
3answers
169 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
0
votes
1answer
65 views

How would I find the residue of $\text{sech}$ and $\coth$ at their poles?

I thought I had understood this, but I'm now lost when trig. functions are introduced and I don't know how to continue. I attempted to apply the $\lim_{z \to a} (z-a)f(z)$ on it, but that didn't take ...
0
votes
1answer
47 views

Integrating Real Function in the Complex Plane

Question: Evaluate the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$ ...
1
vote
1answer
40 views

Calculating $Res_{z=w} {f'(z)\over f(z)} $?

I have $$f(z) = \sum_{n=m}^\infty a_n (z-w)^n $$ where $0 < | z-w | < r$ and $a_m \neq 0$ and am asked to calculate $$Res_{z=w} {f'(z)\over f(z)} $$ I have differentiated $f(z)$ to get ...
0
votes
1answer
337 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
1
vote
2answers
179 views

Integrating $\int_0^{\infty} \frac{dx}{1+x^3}$ using residues.

I want to calculate the integral: $$I \equiv \int_0^{\infty} \frac{dx}{1+x^3}$$ using residue calculus. I'm having trouble coming up with a suitable contour. I tried to take a contour in the shape ...
2
votes
2answers
70 views

Residue/Contour integration problem

Supposedly, $\displaystyle\int_{-\infty}^\infty \frac{\cos ax}{x^4+1}dx=\frac{\pi}{\sqrt{2}}e^{-a/\sqrt{2}}\left(\cos\frac{a}{\sqrt{2}}+\sin\frac{a}{\sqrt{2}}\right)$, $a>0$. Using ...
1
vote
1answer
39 views

Residues of a meromorphic differential on some particular points

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
0
votes
1answer
96 views

real integrals using residues

How to evaluating this integral using residues where $a>0$: $$\int _0^{\infty }\frac{x^3dx}{x^5-a^5}$$ Any help is appreciated
0
votes
2answers
72 views

Calculating $\int_{- \infty}^{\infty} \frac{\sin x dx}{x+i} $

I'm having trouble calculating the integral $$\int_{- \infty}^{\infty} \frac{\sin x}{x+i}dx $$ using residue calculus. I've previously encountered expressions of the form $$\int_{- ...
3
votes
1answer
53 views

Since $A(\alpha)=\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta+\alpha)}{c-i\cdot{d}\cos(\pi+\theta+\alpha)}$, is $A(0)=A(\pi/5)$?

I would like to understand if the result of following integral $$A(\alpha)=\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta+\alpha)}{c-i\cdot{d}\cos(\pi+\theta+\alpha)}$$ is or not ...
0
votes
1answer
37 views

Question on Rudin's Proof of the Residue Theorem

The Theorem in question is Theorem 10.42.: If $f$ is meromorphic in $U$, $A$ is the set of poles of $f$ and $\Gamma$ is a cycle in $U-A$ so that $Ind _{\Gamma}=0$ in $U^c$ then \begin{equation}\frac ...
1
vote
1answer
295 views

Evaluating a Real Improper Integral by Residues

I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as Show that ...
0
votes
1answer
70 views

Calculating $\int_0^{2\pi} \cos^{2n} x \ dx $, please check my work.

In order to calculate the integral $$I \equiv \int_0^{2\pi} \cos^{2n} x \ dx, $$ I first express it in the form $$\int_0^{2\pi} f(e^{it}) ie^{it}dt = \oint_{|z|=1}f(z)dz.$$ By substituting for the ...
2
votes
2answers
43 views

Calculating $\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right)$ with $a$ a double zero of $g$.

I have to show that for $f,g$ analytic on some domain and $a$ a double zero of $g$, we have: $$\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right) = ...
0
votes
0answers
46 views

inverse mellin transform involving $ \zeta(s) $

what is the inverse mellin transform of $ \frac{\zeta (s)}{\zeta (1-s)} $ aplying cauchy theorem only the trivial zeros contribute to the integral so i beleieve that the inverse mellin transform is ...
3
votes
1answer
120 views

Improper integrals are “not totally Improper”

Question is to evaluate $$\int _{-\infty}^{\infty} \frac{dx}{(x^2+a^2)^2}\text {for } a>0$$ Idea is to calculate this using complex analysis/residue theory/contour integration. Approach is ...
1
vote
1answer
145 views

How should I the Residue Theorem to evaluate the integral $\int_{|z|=2} \frac{dz}{(z − 4)(z^3 − 1)}$?

How should I use the Residue Theorem to evaluate the integral $$ \int_{|z|=2}\frac{dz}{(z − 4)(z^3 − 1)}?$$
3
votes
2answers
106 views

How find this sum $\sum_{n=1}^{\infty}\frac{1}{n^2-n+a}$

Today Question if $a>\dfrac{1}{4}$, show that $$\sum_{n=1}^{\infty}\dfrac{1}{n^2-n+a}=\dfrac{\pi}{\sqrt{4a-1}}\cdot\dfrac{e^{\pi\sqrt{4a-1}}-1}{e^{\pi\sqrt{4a-1}}+1}\tag{1}$$ I have konw that ...
2
votes
2answers
52 views

Residues of Complex Functions

I need to find the residues of $f$ at the isolated singular points, namely $z=1,z=0$. Where $f(z)=\dfrac{2z+1}{z(z+1)}$. I already have that the residue at $z=0$ is $1$, and I know I need to do ...
7
votes
3answers
363 views

Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so ...
0
votes
1answer
76 views

Residue of $\frac{1}{(e^z-e)^3}$ at $z = 1$

I'm trying to calculate the residue of $\dfrac{1}{(e^z-e)^3}$ at $z = 1$. The answer is $\dfrac{1}{e^3}$, but having trouble seeing how one would arrive at that. Any hints?
2
votes
1answer
63 views

Integrals of trignometric functions

Question is to Prove that : $$\int_0 ^{2\pi} \frac{d\theta}{a+b\sin \theta}=\frac{2\pi}{\sqrt{a^2-b^2}} \text{for}\ a>b>0$$ using residue theory. What i have done so far is : I transformed ...
2
votes
2answers
99 views

Complex contour integral with residue theory

I need to calculate the following contour integral using residue theory. $z \in \mathbb{C}$ $f(z)=\exp(-1/z) \sin(1/z)$ $\oint_C f(z) dz$ $C: \left | z \right |=1$ The difficult points I ...
3
votes
2answers
191 views

Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
1
vote
1answer
184 views

Can the direction of Contour Integral be affect to the result of integration?

Now i doing the home work about Residue Integration and i doubt that "Can the direction of Contour Integral be affect to the result of integration?" I mean with the same shape of contour but direction ...
4
votes
2answers
289 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
0
votes
2answers
83 views

Residue at $0$ of $\frac{1-\cos z}{z^4}$ [closed]

How do I calculate $Res(g,0)$ of $g=\frac{1-\cos z}{z^4}$?
2
votes
1answer
540 views

Residue at infinity (complex analysis)

I have trouble with the residue : at $z = \infty$. I tried to solve it at $z=0$ but it turns out that I was wrong while $z=0$ is not a pole. I must solve it at $z=2$ but I'm stuck. Any suggestion ...