Questions on the evaluation of integrals along a locus in the complex plane.
1
vote
2answers
39 views
Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
I'm working through Priestley's Complex Analysis (really good book by the way) and this Ex 20.2:
Evaluate $\int^{\infty}_{0}(1+z^n)^{-1}dz$ round a suitable sector of angle $\frac{2\pi}{n}$ for ...
4
votes
2answers
139 views
+50
Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle ...
2
votes
1answer
29 views
Contour Integral
I have this question:
I'm aware that $e^{iz^2}$ is analytic, and hence $I_R = 0$ by Cauchy's Integral theorem. I'm not really sure what to do from there. Thanks!
2
votes
1answer
38 views
Analyticity implying Constant
Question: $f(z)$ is analytic in $C$ and $Im(f(z))\leq 0$. I want to show that $f(z)$ is a constant.
Approach: I know that if $f$ is analytic on a closed curve then the line integral along that curve ...
3
votes
2answers
60 views
Complex integration help
The integral given is
$$\int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx $$
Ok, so, I've used the upper semi circular contour with the function
$$f(z) = \frac{e^{iz}-1}{z^2}$$
Now the residue I ...
2
votes
1answer
43 views
Change of variables in a complex integral
I want to evaluate this integral using Residue Theorem
$$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$
$$ C : |z| = 1 $$
so I substitute letting $$\ W = z ^ {2 } $$
$$ dw = 2z dz $$
and the ...
0
votes
1answer
41 views
$\exp(i z^2)$ on $R e^{i \theta}$ with $\theta \in [0, \pi /4]$.
Let $\gamma_R : [0, \pi/ 4] \ni \theta \mapsto R e^{i \theta}$. I want to show that
$$
\lim_{R \rightarrow \infty} \int_{\gamma_R} e^{i z^2} dz = 0
$$ for $R > 1$. In order to use Jordan's lemma ...
3
votes
1answer
51 views
Contour integral with branch cut
This is a question based on the method here: http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28V.29_.E2.80.93_the_square_of_the_logarithm
The author chose a contour which ...
1
vote
1answer
58 views
Evaluate $\int_{0}^{2\pi} \sin(\frac{\pi}{6} - 2\text{exp}(i\theta)) d\theta$
In one of my exercise sheets, I am asked to find $\int_{0}^{2\pi}\sin(\frac{\pi}{6}-2\exp(i\theta))d\theta$
This follows a question asking to derive a form of Cauchy's theorem:
...
0
votes
0answers
24 views
Definition: “A contour respects causality”
When doing a contour integral, what does "the contour respects causality" mean?
1
vote
1answer
75 views
Evaluating the following integral:
I am trying to evaluate this integral:
$$\int_{0}^{\infty }\frac{\cos(x)}{1+x^{2}}dx$$
My attempt:
$$\int_0^{\infty}\frac{\cos(x)}{(x+i)(x-i)}dx=1/2 \int_{-\infty}^{\infty} ...
1
vote
1answer
52 views
Mellin transform for sin x
I am trying to find the Mellin transform for $\sin x $, in other words
$\int^{\infty}_0 (\sin x) x^{s-1} \mathrm{d} x $
and I know that the answer is
$\Gamma(s) \sin (\pi s/2)$
from several tables ...
0
votes
1answer
40 views
Contour integral, Cauchy's Integral theorem?
Define $$\oint_C f(z) \overline{dz}= \overline{\oint_C \overline{f(z)}dz}\;.$$ If $P(z)$ is a polynomial and $C$ denotes the circle $|z-a|=R$ (counter-clockwise), show that
$$\oint_C P(z) ...
0
votes
4answers
49 views
Contour Integral help with residue theorem
$$
K = \int_{0}^{\infty}\frac{1}{x^{4}+x^{2}+1}dx
$$
I am supposed to use contour integration to solve this, but I can't even determine the singularities. The denominator doesn't have any that I can ...
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
1answer
42 views
Contour integral of $\displaystyle\int_\gamma \dfrac{1}{(2z+1)(z+3)^2}$
Im a little confused by the following integral question
Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the anti-clockwise direction.
$\displaystyle\int_\gamma ...
0
votes
1answer
52 views
Justification in change of variables
it would be fantastic if anyone could help me with the following problem:
I have the integral
$$\operatorname{Im} \left( \int^\infty_0 e^{it} t^{s-1} \mathrm{d} t\right)$$
and I wish to make the ...
0
votes
2answers
35 views
Cauchy integral formula for $\displaystyle\int_\gamma \dfrac{\sin z }{z^4-16}dz$
I have working through past exam questions and I think I have the hang of the Cauchy integral formula and the extended formula... but am a little stuck with how to work these examples out... and the ...
1
vote
2answers
50 views
Contour Integation $\int_\gamma \frac{\cos^2z}{z^2}$
I have the following question from a past exam paper that I'm not really sure how to evaluate. Any help would be appreciated...
Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the ...
0
votes
1answer
59 views
Compute the contour integral $\int_\gamma\frac{\sin z}{z^4}dz$
Im getting really confused looking at past exam style questions evaluating contour integrals... Can anyone help me in the right direction to solve these..
(i) $\displaystyle\int_\gamma \frac{\sin ...
0
votes
2answers
35 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 ...
1
vote
1answer
28 views
Showing a bound on a contour integral
I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
1
vote
1answer
41 views
Proof that $A^n = \frac{1}{2\pi i} \oint _C z^n \left(z - A \right)^{-1} dz$
I'm working my way through Martin Schechter's 'Principles of Functional Analysis' (2nd ed.) and am trying to understand his proof of the following theorem, given on page 136:
"Let $T:X\to X$ be any ...
0
votes
1answer
27 views
Contour Integration & Integration by Parts
I need to find the value of $\displaystyle \int _0^{2\pi}\sin^2 \left(\frac{-\pi}{6}+3e^{it} \right)dt$.
I figured I could use contour integration and the Cauchy-Goursat theorem to do so.
I ...
3
votes
1answer
69 views
Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.
I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...
1
vote
1answer
16 views
Contour Integrals for positively circular contour
Find the contour integral of $\frac{1}{(z^2+1)^2}$ for the positively oriented circular contour $|z-Ri|=R$, for every positive real number $R>\frac{1}{2}$.
I don't know how to set up the ...
4
votes
1answer
80 views
Evaluating $\int_{0}^{\infty} \frac{2 \cos x \ln x + \pi \sin x}{x^2+4} \ dx$
I want to show that $\displaystyle\int_{0}^{\infty} \frac{2 \cos x \ln x + \pi \sin x}{x^2+4} \ dx = \frac{\pi \ln 2}{2e^{2}}$.
The recommendation is to let $\displaystyle f(z) = \frac{e^{iz} ...
1
vote
2answers
73 views
Complex analysis: contour integration
Evaluate by contour integral:
$$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$
Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
4
votes
1answer
47 views
What is the difference between integrals and contour integrals?
I understand integrals but what are contour integrals?
1
vote
2answers
83 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 ...
1
vote
1answer
70 views
Contour integration to compute $\int_0^\infty \frac{\sin ax}{e^{2\pi x}-1}\,\mathrm dx$
How to show:
$$\int_{0}^{\infty}\frac{\sin ax}{e^{2\pi x}-1}dx=\frac{1}{4}\frac{e^{a}+1}{e^{a}-1}-\frac{1}{2a}$$
integrating $\dfrac{e^{aiz}}{e^{2\pi z}-1}$ round a contour formed by the rectangle ...
1
vote
1answer
38 views
A contour integral problem from trinity
A function $\phi(z)$ is zero when $z=0$, and is real when $x$ is real, and is analytic when $|z| \leq 1$; if $f(x,y)$ is the coefficient of $i$ in $\phi(x+iy)$, prove that if $-1<x<1$,
...
1
vote
1answer
58 views
Contour question in complex analysis
Let $C_1$ be the line segment from $-1-i$ to $3-i$, and $C_2$ be the portion of the parabola $x=y^2+2y$ joining the above points $-1-i$ and $3-i$. Show that
$$\int_{C_1}zdz=\int_{C_2}zdz=4+2i.$$
So ...
5
votes
3answers
79 views
Evaluating $ \int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \ dx$
I'm curious about the proper way to evaluate $\displaystyle\int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \ dx = \text{Re} \int_{\infty}^{\infty} \frac{e^{i(x- ...
2
votes
1answer
65 views
Contour integration - Branch cut
I'm asked to show the following equality given $a\in (-1,1)\subset\Bbb R$
$$\int\limits_0^\infty\frac{x^a\ \log(x)}{(1+x)^2}dx=\frac{\pi\sin(\pi a)-a\pi^2\cos(\pi a)}{\sin^2(a\pi)}$$
So I'm trying ...
1
vote
1answer
47 views
Can Cauchy theorem be applied to $\log{(z)}e^{ixz}$?
I'm reading about asymptotic analysis on the integral $I(x)=\int_0^1{\ln{t}e^{ixt}}dt$. The book tells me that I can use Cauchy theorem to deform the contour into a rectangular contour:0->iT, ...
0
votes
1answer
56 views
Calculation of the Inverse Laplace Transform of $\frac{1}{p}$ by contour integration.
I am always told in my lessons of control engineering that the inverse Laplace Transform of $\frac{1}{p}$ is the Heaviside step function $\theta(t)$. But I have a problem when I calculate the inverse ...
1
vote
1answer
47 views
Continuous function on simple closed contour
Let $f$ denote a function that is continuous on a simple closed contour $C$. Using the Cauchy Integral formula, prove that the function $g(z)=\frac{1}{2\pi i}$ $\int_C$ $\frac{f(s)ds}{s-z}$ is ...
0
votes
1answer
34 views
Contour Integrals and positively oriented circles
If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{
\begin{array}{lr}
0 & n=\pm1, \pm2, ...\\
2\pi i & n=0\\
...
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 ...
5
votes
4answers
318 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
1
vote
1answer
85 views
half-line Fourier transform of $x^{z-1}$ w.r.t. $x$?
Can someone help me evaluate $G_g(z)=\int_0^{\infty}x^{z-1}e^{igx}dx$, where $g$ is real and $z$ is complex?
By closing the contour in the upper half plane, I've managed to prove that if ...
0
votes
0answers
70 views
About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$
Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
0
votes
1answer
24 views
Contour integral redefining variables
I have the integral
${\operatorname{Im}} \left (\int^\infty_0 e^{ix} x^{s-1} \, \mathrm{d} x \right)$
and I wish to redefine $x \to iy$ but I am unsure of how to justify this using contour ...
2
votes
2answers
45 views
When Cauchy integral and when Cauchy residue..?
$\int_C\tan(z)dz$ where $C$ is the circle $\vert z\vert=2$
What should be applied to evaluate the following solution?
Is it Cauchy integral or residue?
0
votes
0answers
51 views
About the inverse laplace transform of sinc function
How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ?
Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ .
Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
3
votes
0answers
81 views
contour integration around a dogbone/dumbbell contour
I'm getting the correct answer, but I'm not confident in what I'm doing.
I want to evaluate $\displaystyle\int_{0}^{1} \frac{1}{\sqrt[3]{x^{2}-x^{3}}} \ dx $ using contour integration.
I'm going to ...
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 ...
2
votes
2answers
25 views
$\int_{C^+(0,R)} \frac {dz} {(z^2-1)\dots(z^2-100)}$ independent of $R$ if $R>10$
How can I show that $$\int\limits_{C^+(0,R)} \frac {dz} {(z^2-1)\dots(z^2-100)}$$ is independent of $R$ if $R>10$ without calculating all the residues?
5
votes
2answers
179 views
Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$
How would one compute the following improper integral:
$$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$
without any knowledge of Fresnel equations?
I was thinking of using ...
