Tagged Questions
1
vote
2answers
43 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
154 views
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
47 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 ...
3
votes
1answer
53 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
60 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:
...
1
vote
1answer
76 views
Evaluating the following integral:
I am trying to evaluate this integral:
$$\int_{0}^{\infty }\frac{\cos(x)}{1+x^{2}}dx$$
My attempt:
$$\int_0^{\infty}\frac{\cos(x)}{(x+i)(x-i)}dx=1/2 \int_{-\infty}^{\infty} ...
1
vote
1answer
60 views
Mellin transform for sin x
I am trying to find the Mellin transform for $\sin x $, in other words
$\int^{\infty}_0 (\sin x) x^{s-1} \mathrm{d} x $
and I know that the answer is
$\Gamma(s) \sin (\pi s/2)$
from several tables ...
0
votes
1answer
40 views
Contour integral, Cauchy's Integral theorem?
Define $$\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) ...
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
36 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
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 ...
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
71 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 ...
4
votes
1answer
81 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
76 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
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 ...
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
80 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- ...
1
vote
1answer
48 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
61 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
49 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 ...
2
votes
2answers
106 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
322 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
89 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 ...
2
votes
2answers
46 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?
3
votes
0answers
84 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
83 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?
3
votes
0answers
72 views
Evaluating the integral $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx$
I don't know how to deal with this integral:
$$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx,$$
where z is a complex number.
2
votes
1answer
151 views
Inverse Laplace transform using contour integration
I want to show by contour integration that $\displaystyle\mathcal{L}^{-1} \{\text{arccot}(s) \}(t)= \frac{\sin t\ }{t}$.
In other words, I want to evaluate $\displaystyle \frac{1}{2 \pi i} \int_{a - ...
1
vote
2answers
54 views
Calculating $\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz$ and $\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$
I want to calculate $$\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz\quad\mbox{and}\quad\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$$ using complex integration. In the first part $\sqrt z$ denotes the ...
0
votes
0answers
61 views
Contour Integrals and counterclockwise
$\int_C (z-z_0)^{(n-1)}\ dz$ for any integer $n$, where $C$ is the contour once around the circle $|z-z_0|=1$ counterclockwise and $z_0$ is any point in the plane. Also give the values of the ...
2
votes
1answer
56 views
Evaluate $\int_C\frac{dw}{e^w-1}$ over some loop C contained in the annulus $0<|z|<2\pi$.
Evaluate $\int_C\frac{dw}{e^w-1}$ (counterclockwise) over some loop C contained in the annulus $0<|z|<2\pi$.
Considering the coefficient of $1/z$ in the Laurent series for $\frac{1}{e^z-1}$ by ...
0
votes
1answer
66 views
Contour Integrals
Evaluate:
$\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$.
$\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$
$\int_C (z-z_0)^{n-1}dz $ for any integer $n$, ...
7
votes
2answers
287 views
Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$
I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that
$$\int_0^1 \log(\sin \pi x)dx=-\log2$$
A hint is given as "Use the ...
6
votes
0answers
161 views
Integrate $\ln(x^2+1)/(x^2+1)$ [duplicate]
How to evaluate $$\int_0^\infty \frac{\ln(x^2+1)}{x^2+1} \mathrm{d}x$$ using complex analysis?
I've spent ages trying to think of some clever contour integral which will give it, but I can't seem to ...
1
vote
2answers
66 views
Example of Improper integral in complex analysis
I'm doing this example of Cauchy principle value
$$ \int_0^\infty \frac{dx}{x^3+1}=\frac{2\pi}{3\sqrt{3}} $$
After some steps i got,
$$ \int_{[0,R]+C_R} \frac{dz}{z^3+1}=2\pi i(B_1)\text{
where ...
2
votes
0answers
51 views
Why is contour integration defined only for continuous functions?
Why is contour integration defined only for continuous functions?
Ordinary Riemann integration has no such stipulation (for eg. the Thomae function is discontinuous yet integrable.)
0
votes
2answers
29 views
Simple contour intergal along a circular path
Am I correct that the the following integral evaluates to 0 since the domain of integration is a closed loop and the integrand is continuous over the loop?
$$\int_{C(0,7)}\frac 1{(z-1)(z-3)} dz$$
1
vote
1answer
184 views
Laplace transform of Bessel function of the first kind
I can't figure out why my evaluation of $\displaystyle \int_{0}^{\infty} J_{n}(bx) e^{-ax} \ dx \ (a,b >0, \ n=0,1,2, \ldots)$ if off by a factor of $b$.
$ \displaystyle\int_{0}^{\infty} J_{n}(bx) ...
0
votes
1answer
58 views
Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane
Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ ...
1
vote
1answer
49 views
another Fresnel-like integral
$\displaystyle \int_{0}^{\infty} \sin \left(ax^{2}-\frac{b}{x^{2}} \right) \ dx $
Maple returns numerical results for different values of the parameters that don't agree at all with my answer, and ...
