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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7
votes
1answer
139 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
213 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
137 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
91 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
0answers
96 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
147 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
416 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
117 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
330 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
63 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
37 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
104 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
84 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
44 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
137 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
135 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
36 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
51 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
227 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
135 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
136 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
142 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
181 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
69 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
48 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
349 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
185 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
72 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
97 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
73 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
38 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
299 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
48 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
126 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
146 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
107 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
367 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 ...