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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1
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1answer
25 views

Is there a faster way to compute the residues of this function,

$$\frac{z^2log(z)}{1+z^4}$$ I have that this function has simple poles at $$e^{i\pi/4},e^{i3\pi/4},e^{i5\pi/4},e^{i7\pi/4}$$ which are the zeroes of the denominator (1+$z^4$). The computation of ...
0
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2answers
48 views

Contour integration of exponential function [on hold]

How to solve this integral with residues method? $$\int_0^\infty \frac{e^{ixp}}{x^2+1+i}dx$$
1
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0answers
21 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
4
votes
2answers
52 views

$\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues

I'm trying to find $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx$ where $\alpha>0$ is real. My approach was to take an integral along the real line from $1/R$ to $R$, around the circle ...
3
votes
2answers
50 views

Is there a simpler way to compute the residue of a function at a pole of order 3?

The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the ...
3
votes
1answer
98 views

Computing residues of $\cot(\pi z)/z(z+1)$ with symmetries

I would like to know if there is a quick way of computing the residues of $$f(z) = \frac{\cot \pi z}{z(z+1)}$$at the points $z = 0$ and $z = -1$. They are double poles. Expanding this in Laurent ...
3
votes
2answers
37 views

Help evaluating residue with simple poles

I am having a bit of trouble evaluating $$\sum_{k=1}^3{ \rm Res}\left(\frac{\log(z)}{z^3+8};z_k\right)$$ where $z_1=2e^{i\pi}$, $z_2=2e^{i\pi/3}$ and $z_3=2e^{i5\pi/3}$. I know that each $z_k$ is a ...
3
votes
2answers
462 views

Compute this integral, using a method other than the Residue Theorem,

$\int_0^\infty$ $\frac{1}{1+x}$$\frac{dx}{\sqrt{x}}$ Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$. Part (b) asks, "can ...
2
votes
1answer
29 views

What does complexification mean in complex analysis, .e.g., in residue calculus,

I've learned complexification formally in a graduate linear algebra class. But what does the word mean in the setting of complex analysis? If I consider a real integral on the positive half line, ...
6
votes
1answer
114 views

A particular integral: $\int_{-\infty}^{+\infty}\frac{\sin(\pi x)}{\prod_{k=-n}^{n}(x-k)}\,dx$

I have to show summability, then compute the following integral: $$\int\limits_{-\infty}^{+\infty} \frac{\sin(\pi\,x)}{\prod_{k = - n}^n (x - k)}\,dx = \frac{(-4)^n}{(2\,n)!}\,\pi $$ for every $n\in ...
2
votes
0answers
40 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
1
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0answers
36 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
0
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0answers
27 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
1
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1answer
37 views

Find the residue of $e^{\frac{1}{z^2-1}}\sin(\pi z)$ at $z=1$

I'm dealing with the following problem (from an old qualifying exam): Let $\gamma$ be a closed curve in the right half-plane that has index $N$ with respect to the point 1. Find $$ ...
0
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1answer
38 views

How can I calculate the singularities and residues of…?

$$\frac{e^z}{z^3(z-1)}+\frac{1}{z^3}$$ I have problems specially for $z=0$ Can anyone show me how to do it?
1
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1answer
38 views

Evaluating $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$

Evaluate $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$ where $\gamma$ is the positively oriented boundary of $\{x+iy \in \Bbb{C} : y^2 < (4\pi^2 -1)(1-x^2)\}$. I just learned the residue theorem, ...
7
votes
4answers
220 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
1
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1answer
36 views

Solving $\int_{0}^{+ \infty} \frac{x \cos(x)}{x^4 + 4 a^4} dx$ with residues

We also have the condition $a > 0$. My attempt was to, as usual, define $f(z) = \displaystyle\frac{z e^{iz}}{z^4 + 4 a^4}$. Then I tried to integrate $f$ over a curve $\gamma$ which goes from $0$ ...
1
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3answers
105 views

Solve integrals using residue theorem? [closed]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
0
votes
1answer
36 views

Residue of $\frac{e^{iz}}{z^2+4z+5}$ [closed]

I need to find the residue of $\dfrac{e^{iz}}{z^2+4z+5}$ at its singular points. How do I do that?
2
votes
0answers
61 views

Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
2
votes
1answer
70 views

Computing the residue of $\frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$ for $z = 1$.

Consider the function $$f(z) = \frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$$ We have that $0$ is a double pole and $1$ is a single pole (essential singularity) of $f$. It is simple to compute ...
2
votes
1answer
20 views

Finding the residue, $z=n\pi$, and $e^{n\pi}$

I have reached the following point in a residue calculation and am now unsure what to do: $$Res_{z= n\pi}=\lim_{z\to n\pi}\{(z-n\pi)\frac{ e^z}{\sin(z) } \}$$ $$=\lim_{z\to ...
4
votes
2answers
171 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
2
votes
2answers
41 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
1
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1answer
67 views

Poles on the curve

Say I have this integral: $$\oint_\gamma f(z)\,{\rm d}z,$$and $f$ has a pole on $\gamma$. I understand that we "cut around" the pole with an arc of radius $\epsilon$ and then make $\epsilon \to 0$. ...
0
votes
1answer
30 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
1
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1answer
19 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation ...
0
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2answers
41 views

Computing the residue of $\phi/\psi$ given conditions.

Let $\phi$ and $\psi$ be holomorphic functions around $z = a$, where $\phi(a) \neq 0$ and $a$ is a double root of $\psi(z) = 0$. Prove that the residue of $\phi(z)/\psi(z)$ at $z = a$ is: ...
1
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1answer
69 views

What enclosure should I choose to evaluate the next integral using residues?

I have to solve the next integral: $$\int_{-\infty}^{\infty} e^{ibx}(e^{ia/x}-1)dx$$ where $a,b$ are real parameters. I can use Jordan´s Theorem to show that as $f(z)=e^{ibz}g(z)$ where $g(z)=(e^{ ia ...
0
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1answer
19 views

Residues and poles show that

Show that i) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{z-\sinh z}{z^2\sinh z}=\frac{i}{\pi}$ ii) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{\exp(zt)}{\sinh z}+ ...
1
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0answers
62 views

Using residue theorem find $\int_{\theta-\pi}^{\theta-2\pi}1/(r^2+1-2r\cos u)$

Find the value of $I$, using residue theorem. $$ I= \int_{\theta-\pi}^{\theta-2\pi} \frac{du}{r^2+1-2r\cos(u)} - \int_{\theta}^{\theta-\pi} \frac{du}{r^2+1-2r\cos(u)} $$ with $r<1$ and ...
5
votes
0answers
70 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
1
vote
1answer
38 views

Evaluating residua and simplifying complex expressions.

My question is in two parts, so please forgive its long-winded nature. Lets say that I want to find the residua of the following complex function: $$f(w)=\frac{2w+1}{w(w^3-5)}$$ Let us, ...
2
votes
1answer
56 views

Find the poles/residues of $f(z)=\frac{\sin(z)}{z^4}$

I'm trying to calculate the residua of the following complex function but am encountering problems trying to determine its poles: $$f(z)=\frac{\sin(z)}{z^4}$$ Expanding the denominator shows that we ...
1
vote
1answer
68 views

Evaluation of Residua

Suppose that I have the following complex valued function, and want to evaluate its residua: $$h(z)=\frac {z^5}{(z-3)(z^4+2)}$$ For both parts of the denominator we will have simple poles. For our ...
0
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0answers
50 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
0
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1answer
34 views

Find the residue of $e^{1/z}\sin(z)$ at $z=0$

I am looking for the residues of $f(z)=e^{1/z}\sin(z)$ at its singular points. Found that only $z=0$ is an essential singularity, where $$a_{-1}=\sum\limits_{n=1}^{\infty} ...
1
vote
1answer
45 views

How to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$

I want to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$ (only the first couple of terms). Now I know that if $\sum_{n=-\infty}^{\infty}c_nz^n$ then ...
2
votes
2answers
41 views

Calculating residue $\int_C \frac{8-z}{z(4-z)}dz$

I want to calculate the following: $$\int_C \frac{8-z}{z(4-z)}dz$$ $C$ is a circle of radius $7$, centered at the origin,negative oriented. I want to do this via finding the residues at $z=0,4$. I ...
1
vote
1answer
27 views

Exponential function and residues

Show that singular point of function $f(z)=\frac{1-e^{2z}}{z^4}$ is a pole and find the order $m$ of that pole and the corresponding residue. Here my question begins with the singular point ...
2
votes
1answer
30 views

Singular points and residues

In each case write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular point. ...
3
votes
5answers
276 views

Evaluate the complex integral of function

Use the residue theorem to evaluate $\int_\gamma \frac{z^5}{1-z^3}dz$ where $\gamma$ is the circle $|z|=2$. I have that $z_0=1$ is a singularity point and taking $g(z)=z^5$ and $h(z)=1-z^3$ and ...
0
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1answer
50 views

Residue of $f(z) = \frac{z}{1-\cos(z)}$ at $z=2k\pi$

I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$. I know residue at $z=0$ from here. I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat ...
4
votes
3answers
112 views

Evaluating $\int_{-\infty}^{\infty}\frac{\cos x}{e^x + e^{-x}}$ using the Residue Theorem

I consider the complexification $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Poles of $f$: $\text{Denominator}=e^{-z}(e^{2z}+1)=0\Rightarrow e^{2z}=-1=e^{i(\pi + 2\pi k)}\Rightarrow z=\frac{i\pi(1+ 2k)}{2}$, ...
0
votes
0answers
55 views

Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.

Suppose that $h$ has a pole of order $m\in\mathbb{N}$ at the point $z_0$. Show that there exists a neighbourhood of $z_0$ and a complex differentiable function $g$, so that $g(z_0)=z_0$, where ...
2
votes
2answers
82 views

Calculate the residue of this function

Find the residue at $z=0$ of the function $f(z)=\frac{\cot z}{z^4}$ I know that $z_0=0$ is a pole of order $k=5$, and $$Res(f;z_0)=\frac{\phi(z_0)^{(k-1)}}{(k-1)!}$$ but I cannot get the right ...
6
votes
1answer
72 views

Evaluating $\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$ using Complex Analysis

I am trying to use the residue theorem to evaluate $$I=\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$$ I'll explain my difficulty in finding a contour, then I explain my difficulty in finding a new contour ...
1
vote
3answers
55 views

Complex Integration, residues

Evaluate the following integrals by the method of residues i)$\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$, a real ii)$\int_0^\infty \frac{x^\frac{1}{3}}{1+x^2}dx$ I'm a little lost to ...
1
vote
1answer
49 views

ML-inequality: How to show that $e^{i2x} = e^{i2z}$ when evaluating $\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$

I am to solve the following integral: $$\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$$ We use contour integration in combination with residue calculus, so for $R > 1$ ($R$ is the radius ...