Questions on the evaluation of integrals using the method of residues or in the method's theory.
0
votes
2answers
35 views
Computing real integrals using the Residue Theorem where singularities are on the real line
How would you compute, for $a>0$ the integral $$\int_0^\infty \frac{\sin x}{x(x^2 + a^2)} dx \, \, ?$$
I've computed the residues of the function $$f(z) = \frac{e^{iz}}{z(z^2 + a^2)} $$ which I ...
3
votes
3answers
27 views
Laurent Series and residue of $\frac{z}{(z-1)(z-3)}$ around z = 3
As mentionned in the title, I'd like to get the function's Laurent series and after its residue, I have tried to separate the two denominators to get a partial fraction but I still have a z at ...
3
votes
4answers
75 views
Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?
How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
2
votes
1answer
20 views
Residue of a 1-form in a Riemann Surface does not depend of the chart
Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by
$$
...
2
votes
0answers
32 views
Find the inverse laplace transform: $\frac{1}{{{{({s^2} + 1)}^3}}}$
Find the inverse Laplace transform: $$x(t) = {L^{ - 1}}\left[ {\frac{1}{{{{({s^2} + 1)}^3}}}} \right]$$
with $x(t=0)=0$.
I did:
$${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
4
votes
2answers
55 views
Calculating integral with branch cut.
I'm learning how to calculate integrals with branch points using branch cut. For example:
$$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$
where ...
2
votes
1answer
57 views
is this trig integral doable using contour integration?.
Is it possible to evaluate $\displaystyle \int_{0}^{\pi}\frac{x\cos(x)}{1+\sin^{2}(x)}dx=\frac{-{\pi}^{2}}{4}+ln^{2}(\sqrt{2}-1)$ by using residues?.
I attempted it by considering $\displaystyle ...
5
votes
1answer
66 views
Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$
$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
1
vote
2answers
49 views
Integration using residues
For the following problem from Brown and Churchill's Complex Variables, 8ed., section 84
Show that
$$ \int_0^\infty\frac{\cos(ax) - \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$
where $a$ and ...
1
vote
2answers
36 views
How do I find the residue of a function with a huge exponent?
How would I find the remainder of a function that has a huge exponent that would take ages to work out?
Say I have something like this:
$\frac{5x^{110} + x^4 - 7x^2 - 6}{x-1}$
I honestly don't know ...
1
vote
2answers
48 views
determining residue for the purposes of calculating an integral
Determine the integral
$$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$
using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications.
In order to do this. We ...
2
votes
0answers
43 views
Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$
I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it.
...
0
votes
2answers
36 views
Contour Integral of $\int \frac{a^z}{z^2}\,dz$.
My task is to show $$\int_{c-i\infty}^{c+i\infty}\frac{a^z}{z^2}\,dz=\begin{cases}\log a &:a\geq1\\ 0 &: 0<a<1\end{cases},\qquad c>0.$$So, I formed the contour consisting of a ...
0
votes
3answers
75 views
Cauchy principal value of $\int_{\infty}^{-\infty}e^{-ax^2}\cos(2abx) \,dx$
How do I find out the Cauchy Principal value of $\int_{-\infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx\,\,\,\,\,\,\,\,a,b>0$ using complex integration? The answer is $\sqrt{\frac{\pi}{a}}e^{-ab^2}$, and ...
1
vote
2answers
133 views
$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ using residues
How do I find the following integral by converting it into a complex integral and then using residue theorem?
$$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$$
My approach is as ...
1
vote
1answer
60 views
Sum of residues.
Let $q$ be a polynomial of degree $n$ with distinct zeroes $z_1,\ldots,z_n$. Let $p$ be a polynomial of degree $n-2$ or less. Show that:
$\displaystyle\sum_{K=1}^{n} ...
1
vote
0answers
49 views
Complex Integral using Residues
This is the question:
Find the integral using residue theorem.
$$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta} $$
I solved it like this :
$$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta}=\int_0^{2 \pi} ...
3
votes
1answer
72 views
Evaluating $\int_{C}\left({{e^{2z}\over z^2(z^2+2z+2)}+\ln(z-6)+{1\over (z-4)^2 }}\right) dz$.
Question : Evaluate
$$\int_{C}\left({{e^{2z}\over z^2(z^2+2z+2)}+\log(z-6)+{1\over (z-4)^2 }}\right) dz$$
where C is the circle $|z|=3$. State the theorems your have used to evaluate the integral
...
1
vote
0answers
30 views
Please help with evaluating an integral using the Residue Theorem [duplicate]
Use the Residue Theorem to evaluate $\displaystyle\int_{0}^{∞} \frac{\sin^2(x)}{x^2} \, dx$.
Using the trig identity, this is how far I've gotten:
let $F(z)=\dfrac{1}{2}\dfrac{1-(e^{2iz})}{z^2}$, and ...
2
votes
1answer
53 views
complex analysis explanation
Can someone please explain why :
"Residue at a finite point is zero if the function is analytic at that point".
Some explanation going by the definition or Laurent's expansion will be helpful.
2
votes
2answers
61 views
Question on Residues (Complex Analysis)
Let $f(z)$ be given by
$$f(z)={a_0+a_1z+\cdots+a_{n-1}z^{n-1}\over b_0+b_1z+\cdots+b_nz^n}$$
$b_n\neq 0$. Assume that zeroes of denominator are simple. Show that the sum of residues of f(z) at its ...
4
votes
3answers
131 views
Use the Residue Theorem to evaluate the integral:
$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$
I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
2
votes
1answer
63 views
Use the Residue Theorem to evaluate the following integral:
$$\int_{-∞}^{∞} \frac{x^4}{1+x^8} dx$$
I've found the zeros in the upper half plane to be
$$e^{i \pi/8}, e^{i 3 \pi/8}, e^{i 5 \pi/8}, e^{i 7 \pi/8}$$ (right?)
But then the calculation got really ...
2
votes
3answers
72 views
Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.
Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$.
The contour is ...
1
vote
2answers
84 views
Evaluation of the contour integral $\int_\beta \frac{e^z}{e^z-\pi} dz$
Suppose $\beta$ is a loop in the annulus $\{z:10<\left|z\right|<12\}$ that winds $N$ times about the origin in the counterclockwise direction, where $N$ is an integer. Determine the value of ...
0
votes
1answer
62 views
A question about the residue calculus
Suppose I have a convergent definite integral of the form $$\int_{-\infty}^\infty \frac{f(x)}{x^2(e^x-1)}\text{d}x,$$
where $f(x)$ has no poles, and I want to try to evaluate it using the residue ...
1
vote
2answers
60 views
Calculate $ \int_{\mathbb{R}} \frac{dx}{x^4+1}$ using the residues theorem.
Calculate using the residues theorem this integral : $$ \int \limits_{-\infty}^{+\infty} \frac{\mathrm{d}x}{x^4+1}.
$$
First I calculated $\displaystyle \int_{C_r} \frac{\mathrm{d}z}{z^4+1} $, ...
2
votes
2answers
101 views
Summation of series using residues
Let $P(n)$ and $Q(n)$ be polynomials such that $\displaystyle \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}{Q(n)}$ converges conditionally, that is, the degree of $Q(n)$ is exactly 1 degree more than ...
2
votes
1answer
44 views
Real Pole Residue theorem
I've been studying the residue theorem and I've been having a problem understanding the following result seen here (Eq.7.39) which states :
Given a regular function on the real axis, $g$, then ...
1
vote
0answers
65 views
Improper integral equal to -pi with square root and Cauchy principal value
I'd like to know if the following proof for the value of $I$ is correct, and if there is a simpler solution to it. Also, I will probably encounter more improper integrals like this in the future, and ...
3
votes
3answers
136 views
Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$
Compute the integral:
\begin{equation}
\int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx
\end{equation}
for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...
6
votes
3answers
64 views
Residue Formula application
Using the Residue formula, I've been trying to prove $$\int_0^{2\pi}\frac{1}{a^2\cos^2\theta+b^2\sin^2\theta}\,d\theta=\frac{2\pi}{ab},\quad\quad a,b\in\Bbb R.$$First, it seems like the formula should ...
0
votes
1answer
80 views
Contour integral $\int_{|z|=1}\exp(1/z)\sin(1/z)dz$
Evaluate the contour integral $$\int_{|z|=1}\exp(1/z)\sin(1/z)dz$$ along the circle $|z|=1$ counterclockwise once.
The singularities are $\dfrac1{\pi k},k\in\mathbb{Z}$ plus the limit point $0$. So I ...
3
votes
1answer
51 views
Evaluating the (complex) integral $\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$ using residues.
I am trying to evaluate the following integral.
$$\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$$
where $\gamma$ is the path $\cos(t)+2i\sin(t)$ for $0\leq t <4\pi$.
So, $\gamma$ is an ellipse ...
1
vote
1answer
57 views
Algebraic Properties of Residues in Complex Analysis
I'm interested in how residue at a point operation complies with algebraic operations:
$$\underset{z_0}{\operatorname{Res}}(f + g) = \, ?$$
$$\underset{z_0}{\operatorname{Res}}(f g) = \, ?$$
my ...
1
vote
0answers
71 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
10
votes
1answer
197 views
Compute the inverse Laplace transform of $e^{-\sqrt{z}}$
I want to compute the inverse Laplace transform of a function
$$
F(z) = e^{-\sqrt{z}}.
$$
This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
1
vote
2answers
157 views
Determine and classify all singular points
Determine and find residues for all singular points $z\in \mathbb{C}$ for
(i) $\frac{1}{z\sin(2z)}$
(ii) $\frac{1}{1-e^{-z}}$
Note: I have worked out (i), but (ii) seems still not easy.
5
votes
1answer
119 views
use residues to evaluate sum involving square of csch
I have been trying to evaluate the following sum using residues
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\pi n)}=\frac{1}{6}-\frac{1}{2\pi}$
I am mainly interested in using residues to ...
0
votes
0answers
50 views
Compute a Real Integral using Residue Theorem
My question is related to the computation of the integral:
$$I(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{-iz~p}}{p^2}dp,$$
where $z\in \mathbb R$.
Using the Residue Theorem I found the ...
3
votes
1answer
54 views
Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$?
Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$? If so, is it not meaningful to discuss its residue at $0$?
1
vote
1answer
69 views
Locate poles and calculate residue
Let $a = \sqrt{\pi}e^{\pi i/4} = (1+i)\sqrt{\pi/2} $
Consider the function
$$ f(z) = \frac{e^{-z^2}}{1+e^{-2az}} $$
I have already shown that $f(z) - f(z+a) = e^{-z^2}$ if that helps at all.
The ...
0
votes
1answer
30 views
Residue Theorem for a self-intersecting closed curve?
What does the residue theorem say about a closed curve curve as shown in this figure: figure
It seems to me that this curve self intersect at origin.
It's related to the Wick rotation and I can't ...
2
votes
2answers
72 views
How does $\int_{z=-R+0i}^{R+0i} \frac{e^{2iz}-1-2iz}{z^2}\ dx$ become $\int_{-R}^R \frac{\sin^2x}{x^2}\ dx$?
While trying to compute $\int_0^\infty \frac{\sin^2 x}{x^2}\ dx$, the author of this book suggests computing $\int_{C_R} \frac{e^{2iz}-1-2iz}{z^2}\ dz$ on a semi-circular contour in the upper ...
0
votes
1answer
53 views
Residue Theorem, Find the sum of residues when z is an integer
We have the function
$$ f(z)=\frac{\pi \cot(\pi z)}{(u + z)^2} $$
I already found the residue at the pole when $z = -u$. However there are more poles when z is an integer. How do I go about finding ...
0
votes
2answers
30 views
$\textrm{Res}\left(\frac{\log z}{z^3+8}; z_k\right) = \frac{-z_k \log z_k}{24}$ when $z_k$ solves $z^3+8=0$
The problem in the book is
Compute $\int_0^\infty \frac{dx}{x^3+8}$.
I set up the keyhole contour, apply the residue theorem, and go through the tedious algebra. I get stuck in doing so, but ...
4
votes
1answer
71 views
Strategy for Improper Integrals Related to the Beta Function 2
I am looking for the solution of the following integral
$$\int_0^1 y^k \log\left(1+\left(\frac y{1-y}\right)^a\right)dy,\quad a>0 $$
I really appreciate it if any one can help.
1
vote
1answer
59 views
Using residue theorem separately for real and imaginary parts
I'm trying to calculate an integral with respect to a complex value. I just want to know if I can estimate the integral using the residue theorem separately for the real and imaginary parts of the ...
3
votes
2answers
88 views
A problem on Residue Theorem
Today I had a problem in my test which said
Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$.
Now, clearly this was a misprint since C is not a circle. I tried to find ...
0
votes
1answer
62 views
Sum of all the residues of the function $a(z)/b(z)$
Let $a(z)$ and $b(z)$ be polynomials such that
$ \deg(b) \ge \deg(a)+2$.
Find the sum of all the residues of the function $a(z)/b(z)$.
In class, I learned that
$$
- \text{ sum of all residues of ...
