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

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4 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 ...
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36 views

Show $\int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos x)}{\tan(x)}dx=\frac{\pi^2}{4}$

Yesterday I found this integral on Quora: How does one prove the following integral? $$ \int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos ...
6
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0answers
42 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 $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
7
votes
2answers
87 views

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

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 ...
3
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1answer
87 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
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0answers
13 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 ...
5
votes
2answers
103 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 ...
5
votes
5answers
99 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
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0answers
22 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 ...
3
votes
1answer
45 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
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1answer
42 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 ...
0
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1answer
46 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
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2answers
54 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
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1answer
17 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 ...
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1answer
24 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?
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0answers
34 views

How to prove that the integration contour of one integral is the subset of the other integral? [closed]

I have the two integrals which have the same non negative integrand. For example the following two integrals, $\int_{-l}^{l}f(x)dx$ ....... (1) and $\int_{a}^{b}f(x)dx$ ....... (2) What we ...
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1answer
34 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 ...
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2answers
60 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
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1answer
28 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} ...
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2answers
41 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
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1answer
22 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
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1answer
35 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 $s=0$. ...
1
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1answer
54 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
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1answer
63 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.
4
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1answer
83 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
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1answer
39 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
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1answer
26 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
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1answer
69 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
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2answers
55 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 ...
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0answers
27 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 ...
2
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0answers
39 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 ...
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0answers
23 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.
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32 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 $z(t)=t+it$ ...
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1answer
27 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 $c>0$?
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2answers
78 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 ...
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0answers
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How to integrate $\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$, where $C: |z-1|=1$ using Cauchy's formula?

How can evaluate $$\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$$, 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 ...
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2answers
55 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
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0answers
27 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
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1answer
63 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 ...
0
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0answers
26 views

Evaluating an integral (contour integration probably)

Could you please help me to evaluate the following two integrals? I am looking for the solution of $$\int_{}^{} \frac{A-B\cos(kt)}{\left ( C-D\cos(kt) \right )^{3/2}} dt$$ and $$\int_{}^{} ...
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1answer
39 views

Value of the integral $\int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot e^{i\theta}+k_3\cdot e^{j\phi} +z )d\theta d\phi$

I would like to compute the following integral, which arises from some physics problems, where $k_2$, $k_3$ are real, $z$ is in general complex, $$ \int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot ...
1
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0answers
34 views

Integral along real axis by integrating along contours above and below axis [duplicate]

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $\int_{-\infty}^\infty dk_0 ...
4
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0answers
44 views

Solving an integral (using Cauchy contour integral?)

I need to solve this integral: \begin{equation} f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx \end{equation} where $a$ and $b$ are real, positive ...
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2answers
54 views

How to integrate $x\times \frac{\sin(x)}{x^2+a^2}$ from zero to infinity

I am trying to evaluate $\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$. I get $\frac{\pi}{4} \sin(ia)$ using residue theorem. I integrated over the path that goes from -R to R along the real axis and ...
0
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0answers
30 views

Complex Integrals using a contour [duplicate]

Can anyone help me how to prove this integral
7
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1answer
156 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle ...
0
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0answers
52 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
4
votes
2answers
98 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
10
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1answer
231 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
3
votes
2answers
78 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...