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

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7
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65 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
0answers
267 views

Contour integral representation of Confluent Hypergeometric Function

My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$: From ...
5
votes
0answers
48 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 ...
5
votes
0answers
582 views

contour integral with singularity on the contour

I want to compute the following integral $$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$ The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
4
votes
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62 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: ...
4
votes
0answers
30 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
4
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58 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
172 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
4
votes
0answers
615 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
4
votes
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209 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where ...
3
votes
0answers
37 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do i count the residues at this ...
3
votes
0answers
20 views

Integral of Bessel function with Gaussian over a quadratic

I need help with the following integral: $$ \int_{0}^{\infty} \frac{J_0(ax)xe^{-bx^2}}{1-cx^2}dx $$ Where $ J_0(x) $ is a Bessel function of the first kind (of zero order). I've looked up a few ...
3
votes
0answers
54 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 ...
3
votes
0answers
89 views

An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
3
votes
0answers
123 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
3
votes
0answers
80 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
3
votes
0answers
76 views

Branch-point order and Cauchy representation

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. Suppose we have the following representation: ...
3
votes
0answers
144 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
3
votes
0answers
90 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
3
votes
0answers
50 views

Real integrals using Complex integration

What is the criterion for choosing a contour in the complex plane for the evaluation of real integrals of the form $\int_{-\infty}^{\infty}f(x)dx$? For example $\int_{-\infty}^{\infty}\sin{x^2}dx$.
3
votes
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80 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
3
votes
0answers
191 views

difficult integral involving arcsin(x)

I have a difficult integral to compute. I know the result by guessing the answer, but need to know the method of calculation. The integral is $$ \int_{a}^{b}{\rm d}p\,{p \over p^{2} - 2\mu}\, ...
3
votes
0answers
137 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
2
votes
0answers
20 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
2
votes
0answers
47 views

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t - 1)}}{s(s^2 + 3s + ...
2
votes
0answers
31 views

How is half-contour integration possible?

You integrate in a loop around a singularity $z$ and get $2\pi i \text{Res}(z)$. Is there a path of integration such that the result is $\gamma 2\pi i \text{Res}(z)$ with $\gamma\in (0,1)$? If it ...
2
votes
0answers
25 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
2
votes
0answers
26 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
2
votes
0answers
36 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 ...
2
votes
0answers
26 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 ...
2
votes
0answers
66 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$ \int^1_0 dx \frac{\ln(x-a)}{x-b} $$ by turning it into something in terms of dilogarithms. But for certain values of $a$ and ...
2
votes
0answers
63 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
2
votes
0answers
64 views

Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
2
votes
0answers
70 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
2
votes
0answers
101 views

Fresnel integrals from an inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
2
votes
0answers
100 views

Numerical integration of function with singularities

I am currently trying to solve a semi-infinite integral containing a set of singularities lying on the real axis numerically. The process I am using is breaking the integral into small steps $\Delta ...
2
votes
0answers
66 views

Typical Contour Inegral Proof

I'm trying to prove that the following contour integral approaches 0 as R -> $\infty$. How exactly would we go about doing this? $$ \int{\log\left(z^{2} + 1\right) \over 1 + z^{2}}\,{\rm d}z\quad ...
2
votes
0answers
126 views

What is an example of an integral that CANNOT be done without contour integration ? If that exist.

What is an example of an indefinite integral that CANNOT be done without contour integration ? If that exist. Im talking about closed forms for integrals, not numerical methods. Note that there are ...
2
votes
0answers
286 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
2
votes
0answers
166 views

The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
2
votes
0answers
103 views

quick integration notation question

What exactly does the base of this integral sign refer to? $$\int\limits_{[0\to 1]}$$ Is that an arbitrary contour from the point $0$ to the point $1$? (this is complex analysis) ps. Here's the ...
2
votes
0answers
80 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.)
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vote
0answers
21 views

How do I solve this integral with a branch point at z =0?

The integral $\int_{-\infty}^{\infty}e^{\iota\left(k+\iota\delta\right)x^{2}}dx$ can be written as $\int_{-\infty}^{\infty}\frac{e^{\iota\left(k+\iota\delta\right)z}}{\sqrt{z}}dz$. Here, the branch ...
1
vote
0answers
26 views

Question about integral over cos^3(theta) on complex plane

I had an integral of $$\int_{0}^{2\pi}\cos^3(\theta) d\theta$$ The answer came out to be integral over the curve $$\int_{C} \dfrac{(z^2+1)^3}{8iz^4} dz$$ $$=-i* \int_{C}\dfrac{(z^2+1)^3}{8z^4} ...
1
vote
0answers
17 views

Contour Integral Around a Circle of Large Radius

I'm given the function $$f(z)=\frac{(z^2-1)^{1/2}}{z^2+1}$$ where $-\pi < arg(z \pm 1) \leq \pi$ and the only branch cut required is the section $[-1,1]$ of the real axis. I'm required, using ...
1
vote
0answers
29 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
1
vote
0answers
33 views

Contour integration and the square root branch cut

Consider the following equation $$ \int_0^\infty f(\sqrt{x(x-a)}) dx $$ For $a>0$ real and some analytic function $f(z)$ which dies off sufficiently fast for $\Re[z]>0$ and $\Im[z]>0$ so ...
1
vote
0answers
19 views

Winding numbers are continuous: The proof was too easy

There's a question in my complex analysis book: Let $G$ be a region and let $\gamma_0$ and $\gamma_1$ be two closed smooth curves in $G$. Suppose $\gamma_0\sim\gamma_1$ and $\Gamma$ is a homotopy ...
1
vote
0answers
15 views

How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...