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

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2
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2answers
76 views

About the “mixed” form of Gauss and Fresnel integrals

How to integrate the "mixed" form of Gauss and Fresnel integrals as following? $$\int_{-\infty}^{+\infty} {e^{-x^2-ia(x+b)^2} dx} $$ where $a \in R, b \in R$. [EDIT] As Claude Leibovici pointed ...
1
vote
0answers
33 views

Bessel functions and contour integrals

I have a Bessel function $ x^{2}J''+xJ' + (x^{2}+m^{2})J=0 $ Supposing $ J(x) = x^{m}j(x) $ the equation can be reduced to $$ x(j'' + j) + (2m+1)j'=0 $$ My question is, how do i show that $$ ...
0
votes
0answers
16 views

Analyticity of Mellin Barnes integral

How to decide the analyticity of Mellin-Barnes integral? In particular, When Fox's H-function is analytic? Is the condition for existence, analytic and condition for convergence both have the same ...
0
votes
0answers
11 views

Can any one tell m ewith one example, how to evaluate a double Mellin Barnes integral?

What is meant by asymptotic expansion of Gamma function? i.e. $ |\Gamma(z)| = |\Gamma(x+iy)| \approx \sqrt{2 \pi} \left|y\right|^{\left(x - \frac{1}{2}\right)} e^{-\pi \frac{|y|}{2}}, \quad ...
0
votes
1answer
34 views

Contour integration on semicircle as R -> infinity

$$f(z)=\frac{e^{iz}-1-iz}{z^3}$$ What is the value of $$\int_{C} f(z) dz$$ if C is the arc of the semicircle with radius $R\to \infty$ ,going counterclockwise from point (R,0) to (-R,0) Attempt: I ...
4
votes
2answers
224 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: ...
1
vote
3answers
96 views

Contour Integration of Line Segments

I am trying to use contour integration to find the integral of: $$ \int_\gamma ydz $$ where we have the union of line segments from $0$ to $i$ and then to $i+2$. I simply do not understand how to ...
1
vote
1answer
30 views

Integrate cos(z) over a quarter of an ellipse.

The complex form of the equation for an ellipse with foci at 1 and -1 is $|z-1|+|z+1|=\sqrt{8}$. a) Find the values of $a$ and $b$ such that $x^2/a^2+y^2/b^2=1$ describe the same ellipse. b) Let $C$ ...
1
vote
2answers
63 views

$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$

I want to examine the convergence of the series $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$$ In case it converges I want to evaluate it. I tried the D' Alembert theorem but it was ...
2
votes
0answers
32 views

Contour integral with signum function

I need to solve the following integral $$\int\limits_{\left| {s - a} \right| = \delta } {\frac{{{\mathop{\rm sgn}} (is)}}{{{e^{2\pi irs}}}}ds}$$ Where the contour is the semicircle in the upper ...
5
votes
1answer
96 views

A difficult one-variable exponential integral

I am trying to work out a closed form for the integral \begin{equation} \int_{0}^{1} \frac{1}{\sqrt{s(1-s)}} \exp\left(-\left(\frac{a}{s} + \frac{b}{1-s}\right) \right) \,ds \end{equation} where ...
2
votes
1answer
52 views

Is the integration $\int_{C_R}e^{ikz}dz=0\ (\text{if}\ k>0)$ correct?

When I read P.W. Milonni's book "Fast light, slow light, and left-handed light", I encounter this problem. In chapter 2 of the book, the author introduces the integral ...
1
vote
3answers
36 views

What is the length of the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$?

Let $C$ be the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$. What is the length of $C$? Would the length of $C$ be $5$ or $10$? I think $r=5$ so I am not sure whether that would be the ...
1
vote
1answer
77 views

Complex Analysis (Contour Integration)

Given complex numbers $z_1$ and $z_2$, let $[z_1, z_2]$ denote the straight line segment path from $z_1$ to $z_2$. Recall that we can parametrize this by $x(t) = z_1 + t(z_2 - z_1)$ for $t \in ...
2
votes
2answers
121 views

Calculating an integral (using methods from complex analysis) (hints only please)

From Rudin's book, we are to calculate $\int_\mathbb{R} \Big(\frac{\sin x}{x}\Big)^2 e^{itx}dx$ where $i$ is the imaginary number and $t\in\mathbb{R}$. I'm looking for a hint on how to get started. I ...
1
vote
0answers
35 views

Evaluate Complex Line Integral

Evaluate $\int _C f$ where $f(z)=x^2+iy^2$ and where $C$ is given by $z(t)=t^2+it^2, 0\leq t \leq 1$. I tried reading an example in the book, using the formula $\int_C ...
1
vote
1answer
112 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
5
votes
2answers
227 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
2
votes
1answer
42 views

Contour integral over a segment

Let $S$ denote the segment that connects the points $O(0, 0), \; A(1, 1)$. I want to evaluate the integral: $\displaystyle \int_{S} z^2 \, dz$. The segment can be parametrized as $\gamma(t)=(t, t), ...
0
votes
1answer
34 views

integration of an open curve about isolated singularities

I know if I integrate a circular arc of an angle $\theta < 2\pi$ about an isolated singularity of the complex funciton I would get a fraction $\frac{\theta}{2\pi}$ of the residue of that ...
1
vote
1answer
45 views

contour integral in a region where the function doesn't have any poles

What is the value of the following contour integral? The contour is a circle with radius $0.5$ around $z=i$ point: $|z-i|<\frac{1}{2}$ $$\oint_C\frac{dz}{2-\sin z}$$ I think it is $0$ because ...
1
vote
3answers
65 views

How To Find The Length Of An Irregular Arc

How would you find out the length of an irregular arc. e.g. An arc with a base length of $10$cm and a height of $5$cm - what would be the length of that arc? Is there a specific formula I could use?
0
votes
1answer
57 views

How can I calculate the integral $ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $ [closed]

How can I calculate the integral? $$ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $$ for $ \left| a \right| < r < \left| b \right|$ and $ m, n > 1$ I tried to use the cauchy ...
4
votes
3answers
155 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
0
votes
1answer
77 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
votes
2answers
113 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
1
vote
1answer
49 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
3
votes
0answers
41 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
2
votes
0answers
47 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
11
votes
4answers
212 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
5
votes
1answer
38 views

Integral principal value with $\cos$ and $x^2$

Could you tell me how to solve this integral? $$\int_0^{\infty} \frac{\cos x -1}{x^2}dx$$ I think I should focus on this integral $$\int_{\Gamma} \frac{e^{iz}-1}{z^2+ \varepsilon^2}$$ where ...
2
votes
1answer
43 views

Integral with denominator raised to n-th power, residues

I don't know how to calculate this integral: $$\int_{-\infty}^{\infty} \frac{d x}{(1+x^2)^{n+1}}$$ If we denote by $\Gamma$ a curve = semicircle centered at $0$ with radius $R$ + segment $[\ R, R]$, ...
2
votes
2answers
164 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
5
votes
0answers
62 views

Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...
3
votes
1answer
115 views

How does a simple elliptic integral solve this monster?

During some electromagnetics calculation regarding a loop antenna I stumbled across the following integral $$\int_0^{\pi/2} \frac{d\phi}{\big(1+\frac{k}{k-2}\cos(2\phi)\big)^{3/2}}$$ and Mathematica ...
2
votes
1answer
68 views

Complex integral problem - Two different answers! - $\oint_C\frac{dz}{z(2z+1)}$

This is from Arfken, problem #11.4.8 (7th Edition). I have to compute the complex integral, $$\oint_C\frac{dz}{z(2z+1)}$$ over the unit circle. So I took my $f(z)=\frac1{2z+1}$, and my $z_0=0$, and ...
5
votes
1answer
86 views

How to perform this contour integration with $\log$ in the denominator?

Let $k > 0$ and $ a>1$ be constants. As far as I can tell, the integral $$ J = \int_{-\infty}^\infty dx\frac{e^{i k x}}{1+x^2}\frac{1}{\log(a - ix)} $$ converges, since the argument of the ...
8
votes
3answers
182 views

How to prove $\int_0^1 \frac1{1+x^2}\arctan\sqrt{\frac{1-x^2}2}d x=\pi^2/24$?

Since I'm stuck at this final step of the solution here. I wished to try contour integral, taking the contour a quadrant with centre ($0$) and two finite end points of arc at $(1),(i)$: Then: ...
10
votes
1answer
122 views

Integrate $\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$ using residues

I want to find the integral $$I=\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$$ using contour integration; I am familiar with the trigonometric substitution in real analysis. There are no branch cuts, ...
7
votes
2answers
138 views

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

What is the best way to calculate the following sum?$$S=\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi^2}{48}\ln 2$$ I tried putting $$f(z) = ...
1
vote
1answer
72 views

What's the difference betwen parameterizations and variable substitution for solving integrals?

Asumming I have the following integral to solve in the complex plane: $$\int \frac{dz}{z+1} $$ while $|z|=5$ which means a contour of radius 5 around zero. Is it possible to solve this integral using: ...
2
votes
1answer
248 views

Integration of $\ln $ around a keyhole contour

I want to evaluate the following integral: $$\int_{0}^{\infty}\frac{\ln^2 x}{x^2-x+1}{\rm d}x$$ I use the following contour in order to integrate. I considered the function $\displaystyle ...
9
votes
4answers
358 views

Evaluate the integral $\int_0^\infty \frac{x (\ln(x))^2}{x^4 + x^2 + 1}\text{ d}x$

What is the value of $\displaystyle\int_0^\infty \frac{x (\ln(x))^2}{x^4 + x^2 + 1}\text{ d}x$? This is a question I came up with myself. It is not homework. I constructed this example to make the ...
1
vote
1answer
104 views

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods?

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods? In all of the examples I have seen, poles are simple and this helps to avoid ...
4
votes
3answers
126 views

$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $ using contour integration

I am trying to evaluate $$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $$ by using contour integration. It is possible to compute this integral using real techniques; integration by parts yields the ...
2
votes
0answers
69 views

Can this integral similar to the Fourier transformation of $\delta$ function be calculated analytically?

I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$ where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier ...
4
votes
0answers
78 views

Evaluating $\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx$

I was trying to compute: $$ I_{1/2}=\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx. $$ I know it can be recast as follows $$ I_{\alpha}=\int_0^\infty \frac{x^\alpha}{e^x-1}\ dx= \int_0^\infty ...
4
votes
1answer
108 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
2
votes
0answers
71 views

Calculate contour integral (Cauchy integral formula)

I have to calculate (without refering to residue theorem) $$\int_{\partial B(2,3)} \frac{dz}{z^4-16}$$ My attempt: First, I need to find singularities of $f(z)=\frac{1}{z^4-16}$. ...
10
votes
3answers
215 views

Evaluating $\int_{-\infty}^{\infty}\frac{\sin ax-a \sin x}{x^3(x^2+1)} \ dx$ using contour integration

How would you compute the integral $$\int_{-\infty}^\infty \frac{\sin ax-a\sin x}{x^3(x^2+1)} \ dx ?$$ We will integrate along two circular contours and a striaghtline section between them.(Half donut ...