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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3
votes
1answer
15 views

$a_{-1}$ term of a function on the punctured plane?

Our instructor said if f is analytic on the punctured complex plane (missing origin), then $$a_{-1}=\oint_{|z|=r} f(z) dz$$ This may be an obvious question, but what happened to the $\frac{1}{2\pi i}$ ...
2
votes
2answers
64 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
0
votes
1answer
28 views

Residue Calculation Problem

I need to find the residue of the following equation at $z = 0$. $$ \frac{\cot(z)\cot(hz)}{z^3}$$ My attempt is as follows: The residue will be the coefficient of $1/z$ in the Laurent Series ...
0
votes
1answer
126 views

Residue theorem in evaluating complex integrals?

It's been a while since I used residue theorem to evaluate anything. I remember that whenever we have a real valued function, we can use residue theorem to evaluate its integral with associated ...
0
votes
2answers
124 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
1
vote
1answer
44 views

complex residue involving exponent of quotient of polynomials

I was trying to work out an integral and came to trying to find the complex residue of $$R = \exp\left(\frac{Ax^2 + Bix}{Dx + 1}\right)$$ at $x = -D^{-1}$. I used partial fractions to get: $$ = ...
1
vote
1answer
99 views

$\int e^{\cos(x)} \cos(nx)\ dx$ using the residue theorem

I am trying to evaluate the following integral using the residue theorem: $$\int_0^{2\pi} e^{ \cos(\theta)} \cos(n\theta) d\theta$$ I have already evaluated $\int_0^{2\pi} e^{e^{-i\theta}} e^{i ...
5
votes
1answer
274 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
0
votes
0answers
32 views

Contour Integral of $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ [duplicate]

I'm trying to evaluate the following integral: $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ $0<a<1$ I've integrated from 0 to $ i\infty $ then from ...
4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
1
vote
1answer
130 views

Compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ by residues

We want to compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ with $t>0$ using residues. The first thing I want to do is using $z=e^{i\theta}$ to transform the integral ...
1
vote
1answer
121 views

Calculation of a Residue

Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find ...
0
votes
0answers
101 views

Convergence of improper real integrals using the residue theorem

Let $f:\mathbb{R} \to \mathbb{R}$ be a funktion that satisfies $$ |xf(x)| \to 0 \qquad (|x| \to \infty) $$ If $f$ can be extended to a holomorphic function with finitely many singularities $S$ and the ...
1
vote
2answers
108 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
2
votes
0answers
51 views

Integral on the circle

It's a standard fact that to calculate integrals of the form $$\int_{0}^{2\pi}\mathcal{R}(\cos(\theta),\sin(\theta)) \ d\theta$$ with $\mathcal{R(x,y)}$ a rational function in two variables without ...
1
vote
1answer
49 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
3
votes
1answer
153 views

prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...
0
votes
0answers
44 views

Residue at Infinity Question

So, I'm trying to determine a residue of the following at infinity: $$f(z)=\frac{z^6}{(1+z)^3}$$ How would I even begin to approach this? Here are my thoughts; I'm mostly looking for a confirmation ...
0
votes
1answer
75 views

Residue integration

Evaluate\begin{align} \int_{-\infty}^{\infty} \frac{\sin(x)}{x^2+4x+5} \, dx \end{align} I set $f(z) = \frac{1}{z^2 + 4z + 5} = \frac{1}{(z-z_0)(z-\bar{z_o})}$, where $z_0 = -2+i$ and $\bar{z_0} ...
0
votes
3answers
52 views

How to calculate $I(x)=\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ by Residue theorem [$|x|>1$]

How to calculate $\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ for $|x|>1$ by Residue theorem? I could do is just as: $$I(x)=\int_{-\pi/2}^{\pi/2}\frac{d\theta}{\sin\theta-x}\\ ...
0
votes
1answer
180 views

Using Jordan's Lemma for residue integration

Evaluate \begin{align} \int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} \, dx \end{align} So far, I set $f(z) = \frac{1}{z^2+1}$, and the singularity point are $z = \pm i$. But I am using the top ...
2
votes
3answers
904 views

Using residues to evaluate an improper integral

Use residues to evaluate the improper integral \begin{align} \int_{0}^{\infty} \frac{1}{(x^2+1)^2} \, dx \end{align} First, I said $f(z) = \frac{1}{(z^2+1)^2}$. My only question so far is how ...
1
vote
1answer
57 views

Finding a residue

I am given $f(z) = \frac{\text{Log} z}{(z^2 + 1)^2}$. The singularity point occurs at $z = i$, with a pole order of $m = 2$. How can I go about finding $\text{Res}_{z=i} f(z)$? The correct answer ...
0
votes
0answers
43 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
15
votes
3answers
354 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
0
votes
0answers
56 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
0
votes
2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
0
votes
0answers
35 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
7
votes
3answers
175 views

How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

How show that $$\sum_{k=0}^{\infty}\dfrac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\dfrac{\sqrt{2}\pi^2}{16}$$ My idea:I know how to prove the following sum ...
0
votes
1answer
47 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
3
votes
2answers
201 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
4
votes
1answer
143 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
4
votes
0answers
68 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
0
votes
0answers
20 views

A question on Residues and a family of functions

I have the following: $$\alpha(t)=\int_{\left|z\right|=1}z^{k}\frac{f^{\prime}_{t}(z)}{f_{t}(z)}\,dz$$ where $f_{t}(z)$ is a family of entire functions depending on $t \in \Delta$ and $k \geq 0$. By ...
1
vote
3answers
81 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
0
votes
1answer
72 views

Evaluating a few complex integrals on the unit circle

I'm stuck on a few of these, but I have most of the details worked out: (i) $\int_{|r|=1}(z^2-4)^{-1}\,dt=\int_{0}^{2\pi}ie^{i\theta}(e^{2i\theta}-2)^{-1}\,d\theta$ (ii) ...
3
votes
0answers
45 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
2
votes
3answers
128 views

Confusion about a way to compute a residue at a pole

Suppose I have a function of the form $$f(x)=\frac{1}{(x-a)(x-b)^2(x-c)^3}$$ Clearly, I have a simple pole at $a$, and poles of order 2,3 at $b,c$, respectively. By definition, the residue at ...
2
votes
1answer
60 views

$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0$

Let $f \in H(\mathbb{C}- \{ z_{1}, \dots, z_{n} \})$. I need a proof of the fact that $$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0.$$ Where can I find it ?
1
vote
1answer
527 views

What is the residue of $f(z)=\tan{z}$ at any of its pole ? Is the solution correct?

The residue of $$f(z)=\tan{z}$$ at any of its pole is, $$f(z)=\tan{z}=\frac{(z-\frac{\pi}{2})(\tan{z})}{(z-\frac{\pi}{2})}$$ $$\begin{align} \left({\operatorname{Res} {f(z)=\tan{z}; ...
0
votes
2answers
235 views

Prove a function has a removable singularity at $z=0$.

Let $f$ be a holomorphic function on $\mathbb{C}\smallsetminus \{0\}$. Suppose $\int_{|z|=1}z^nf(z)\,dz=0$ for any $n=0,1,2,\ldots$. Prove that $f$ has a removable singularity at $z=0$. How to prove? ...
0
votes
1answer
55 views

Find the residues of $f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}}$

Consider the function $$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$ I want to calculate the residues of $f$ in $\{+i,-i\}$. Using the usual techniques, we have that ...
3
votes
2answers
68 views

Using residue to find a complex integral

Given the following: $$\int_{\varGamma_R} {z\,dz\over e^{2\pi iz^2}-1}, \ \ \ \varGamma_R=\{z\in \Bbb C:|z|=R\},\quad n<R^2<n+1,\,n\in\Bbb N.$$ I want to use the residue for this, but I can't ...
4
votes
1answer
166 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
2
votes
2answers
98 views

About the integral $\int_{0}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}$

I need to prove the following identity: $$\forall a>0,\qquad\int \limits_{-\infty}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}=\pi(1-e^{-a}).$$ I think it can be proven using Laurent series. I tried ...
4
votes
1answer
59 views

Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
17
votes
0answers
484 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
75 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
62 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
108 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 ...