Questions on the evaluation of integrals along a locus in the complex plane.

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2
votes
1answer
34 views

How to calculate the residue of the fourier transform?

I have been struggling calculating the Fourier transform of $f(x)=\frac{x}{(x^2+1)^2}$. I tried to calculate $f(t)=\int\frac{x}{(x^2+1)^2}e^{-ixt}\,dx$ directly by integration by parts, but it is not ...
2
votes
2answers
63 views

Inverse Laplace Transform of e$^{-c \sqrt{s}}/(\sqrt{s}(a - s))$

I am trying to find the Inverse Laplace of the following function: $$ F(s) = \frac{\mathrm{e}^{-x b \sqrt{s}}}{ b (a - s)\sqrt{s}} $$ I really don't know where to start on this one as I have only ...
4
votes
1answer
99 views

Contour integration of $\int_{-\infty}^{\infty}\frac {\sin^3 x}{x^3} dx$: where are the singularities?

I have just begun to study complex analysis and I'm trying to calculate $$ \int_{- \infty}^{\infty} \frac {\sin^3 x}{x^3} dx $$ with the "help" of an exercisebook. I have followed these steps: ...
1
vote
5answers
66 views

Evaluating a contour integral where C is a square

I've been working problems all day so maybe I'm just confusing myself but in oder to do this, I have to the take the integral along each contour $C_1-C_4$ My issue is how to convert to parametric ...
2
votes
1answer
64 views

Evaluate the integral of the given contour

I'm being asked to evaluate $\int \frac{1}{z^3(z^2+1)}dz$, where C is the circle $\lvert z-1 \rvert=\frac32$ I started by determining the zeroes, which are $0, -i, \,i$ Then I applied the Cauchy ...
0
votes
1answer
29 views

Complex definite integral $\int_{0}^{\pi}\frac{ire^{it}}{2-2ire^{it}}dt$

I am trying to evaluate the integral $$\int\limits_{0}^{\pi}\dfrac{ire^{it}}{2-2ire^{it}}dt$$ but I don't know how to proceed.
1
vote
1answer
53 views

Evaluating contour integrals along given C's

Ok, so I have the following problem that I am working on. It says to evaluate $$\int \frac{z}{(z-1)(z-2)}dz$$ where C are given by \begin{align} a)& \ \ C:\lvert z \rvert=\frac12\\ b)& \ \ ...
0
votes
1answer
51 views

Find the value of the integral on the contour C

Ok, so I'm trying to figure out this problem. It asks to find the value of the contour integral $\dfrac{e^z}{z^2(z-\pi i)}$ on the contour $C$ shown in the following figure I believe that in order ...
0
votes
2answers
68 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
0
votes
1answer
26 views

Contour integral of convergent power series

Given that $\frac{e^z}{z^k} = z^{-k} + z^{1-k} + \frac{z^{2-k}}{2!} + \frac{z^{3-k}}{3!} + ...$ converges uniformly on any set $\{z \in C: r \leq |z| \leq Z\}$ (where $0 < r < R$), show that for ...
5
votes
1answer
65 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
0
votes
0answers
26 views

How to show that integration contours are related?

I have one geometry below in which the integration contours are shown with red and blue line. How I can show that the contour in blue line i.e (B to C) is with in the integration contour in red ...
1
vote
1answer
64 views

Inequality complex integral with $|f|\le 1$.

Let $f:\mathbb C\longrightarrow \mathbb R$ be a continuous function such that $\,\lvert\, f(z)\rvert\le 1$ for all $z\in S^1\subset \mathbb C$. Prove that $$\left| \int_{\lvert z\rvert=1} ...
0
votes
0answers
31 views

Integration contour relationship.

We have the two integration contours as shown below, How we can prove that the integration contour B is the subset of the integration contour A? Also note that the figures does not represent the ...
7
votes
0answers
73 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
11
votes
3answers
204 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
12
votes
4answers
545 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
2
votes
0answers
38 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
6
votes
2answers
155 views

Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

Prove using contour integration that $\displaystyle \int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$ I am at a loss at how to start this problem and which contour to pick. I ...
6
votes
7answers
196 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
2
votes
0answers
28 views

contour intrgration, what's the right answer?

There exists an integral as follow: $$ \bar G(t)=\int_{-\infty}^{\infty}\frac{dE}{2\pi\hbar}e^{-iEt/\hbar}\frac{1}{E-\epsilon+i0^{+}} $$ My solution is: $$ {2\pi\hbar}\bar G(t)=-i\pi e^{-i\epsilon ...
4
votes
1answer
62 views

inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
3
votes
1answer
50 views

$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}$

Apparently from Mathematica we have: $$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}=\frac{\pi^2\mathrm{sech}^2\left(\frac{\pi t}{2a}\right)}{4a^2}$$ for $a,t$ both real and positive. I ...
1
vote
1answer
56 views

Use contour integration to calculate real integrals

I tried to prove this equality: $$\int_0^{\pi/2} e^{-x\cos \theta}\ \cos (x\sin \theta)\ d \theta=\frac \pi 2 - \int_0^x \frac {\sin u} u du$$ I calculated $$\int_{\gamma^+} \frac {e^{iz}} z dz$$ ...
3
votes
2answers
60 views

Integral of $\int_0^{2\pi} \frac{e^{-it }dt}{e^{it}-z}$

Sorry if this question seems stupid, but I am confused here: Does it follow that $$ I(z)=\int_0^{2\pi} \frac{e^{-it}dt}{e^{it}-z} = 0 $$ For every $z$ with $|z|<1$? I think this is true. I ...
0
votes
1answer
20 views

Contour Integral About A Circle

I have $\int_\delta \frac{z}{z^3 -1} dz$, where $\delta(t) = (\frac{1}{2})e^{it }$ with $t \in [0, 2\pi]$. It's clear that the winding number is $Ind_\delta(z_0) = 1$, where $z_0 = 0$. I'm just ...
1
vote
1answer
41 views

Parametrization of a Complex Path/Contour Integration

How would I parametrize the path which is a straight line from 1 to a complex point z? Does $\delta (t) = z^t$ make any sense?
0
votes
1answer
39 views

Complex integral with imaginary exponent: $\int_0^\pi i \exp((i\theta)^{1+i}) d\theta$

How to approach the integral $$ \int_0^\pi i e^{(i\theta)^{1+i}} d\theta $$ I know I can't multiply the exponents, but what can I do? Am I at least right that the above is equivalent to $\int_0^\pi ...
1
vote
2answers
65 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
2
votes
1answer
33 views

Complex integration with real integral

If $\gamma$ is unit circles $A(0,1)$ parameterization of one positive rotation and $a\in\mathbb{R}$, $0<a<1$. Show that $$ \int\limits_0^{2\pi} \frac{dt}{1+a^2-2a \cos t}=\oint\limits_{\gamma} ...
-1
votes
2answers
52 views

Complex integration of exponential function

I am asked to find the integral of $z e^{z^2}$. I have applied the formula of multiplication but the factor of exp cannot be eliminated ofcourse. So how can i solve it. Sorry for such a basic question ...
1
vote
1answer
26 views

Using Cauchy's Theorem on Contour Integral

I need to solve $\int_\gamma (1-e^z)^{-1}$ if $\gamma (t) = 2i + e^{it}$. I would assume Cauchy's Integral theorem applies here, where $\gamma$ is a closed path on a convex open set. I'm having ...
1
vote
1answer
54 views

Inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$

What is the inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$? Using the residues, I can calculate the residues at $s_n=2n\pi i$, but I have problem in calculating residue at $s=0$. ...
2
votes
1answer
60 views

Approximating a Gaussian integral

I have been struggling with an approximation to the following integral \begin{equation} \text{p.v.}\int_{-\infty}^{\infty} {e^{-s^2/2v} \over (e^{-2s}- q a)^2} {ds \over \sqrt{2 \pi v}} \end{equation} ...
2
votes
2answers
83 views

Calculating a complex definite improper integral: $I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx$

Does anyone know how to find the value of this integral: $$I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx,$$ where $i=\sqrt{-1}$ and $t$, $a$ are real. Please give me a hint. Thank you.
5
votes
1answer
106 views

A Cosine Integral

What is the value of the Cosine integral \begin{align} \int_{0}^{\infty} \cos\left( \frac{x (x^{2}-a^{2})}{x^{2}-b^{2}} \right) \, \frac{dx}{x^{2} + p^{2}} \, \, \, ? \end{align}
3
votes
1answer
55 views

Integrals over closed contour

Calculate integral $$\oint_{\gamma} \frac{z+1}{z^4+2iz^3} dz$$ where $\gamma$ is parameterization of circle $B(0,1)$ along one positive rotation. I did something like this with Cauchy. \begin{align} ...
0
votes
1answer
27 views

Real value of a complex contour integral equal to the contour integral of the real value of a complex function?

Exactly as in the title: is it generally true that $Re(\int_\gamma f(z)dz) = \int_\gamma Re(f(z))dz$. If not, what would be a case in which it is false? I was thinking a counterexample would follow ...
1
vote
1answer
75 views

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$?

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$? By letting $z = \mathrm{e}^{it}$, we get $$\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz = \int_0^{2\pi} \mathrm{Log}(\mathrm{e}^{it}) ...
0
votes
2answers
59 views

What is the relation between two integrals?

Let us suppose that we have two integrals, $I_1$ and $I_2$ with the same non negative integrand. The integration contour of $I_1$ is a subset of the the integration contour for $I_2$. What can we say ...
0
votes
0answers
28 views

Integration contour as points of set.

If B is the subset of A, I wounder do we have two integration contour or one for this? What will happen if we take AUB i.e. A union B, than do we have one integration contour? and what if we take A ...
3
votes
0answers
60 views

Confused about pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
0
votes
0answers
24 views

How to select the integration contour

In the following two figures which describe sets, How many possible integration contour we have for the figure 1 and how many integration contour we have for figure 2.
1
vote
0answers
36 views

Contour integrals for $f(z)= e^{3z}$

Integrate $f(z)=e^{3z}$ along line segment from $(0,0)\to(1,1)$ parabola $y=x^2$ from $(0,0)\to(1,1)$ circle $|z|=3$ once around its arc (positive $360^o$) First I parametrized with $z(t)=t+it$ ...
1
vote
1answer
36 views

Solve $\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx $

How to prove that $$\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx = 0,$$ where $c>0$?
1
vote
2answers
88 views

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$ May I verify if my solution is correct? Thank you. Consider ...
3
votes
2answers
68 views

How to integrate $\int_C{\frac{\sin\pi z}{(z^2-1)^2}}dz$, where $C: |z-1|=1$ using Cauchy's formula?

How can evaluate $$\int_C{\frac{\sin\pi z}{(z^2-1)^2}}dz$$, where $$C: |z-1|=1$$ by using Cauchy's formula. I have to use Cauchy's formula. Cauchy's formula $$f(z_0)=\frac{1}{2\pi ...
1
vote
2answers
59 views

Evaluate the $I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$

I want to evaluate $$I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$$ It seems that the solution should be in the form of the error function and also it involves contour ...
0
votes
0answers
36 views

Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
1
vote
1answer
73 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...