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

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0
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3answers
65 views

Conversion into contour integral and poles

Say I have this integral $\int_{-\infty}^\infty$ $\frac{x^2}{x^6+1}$ dx . Now I know that it has six poles according to this denominator which are the six roots for -1. The question is after I split ...
1
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1answer
35 views

How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + ...
-1
votes
1answer
17 views

Evaluating a contour-integral.

Consider the ellipse $C$ given by $x^2 + y^2/4 = 1$. How to evaluate $$\int_C x^2 \, \nu(d(x,y))$$ where $\nu$ is the Lebesgue length measure on $C$? I am not sure if this can be computed like a ...
2
votes
2answers
63 views

What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
3
votes
0answers
73 views

Contour Integral of Square root Function. Branch Cuts

I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ...
3
votes
0answers
40 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...
1
vote
0answers
54 views

Find the different values of this integral when all paths of integration are possible

This is the question.. I only know how to do the question from infinity to infinity.. enter image description here Find the different values of this integral when all paths of integration are ...
2
votes
1answer
75 views

How do I evaluate the following Integral

The integral is $$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$ where $m, a$ and $b$ are real numbers such that the integral is definitely convergent. Any ideas on how to solve ...
1
vote
2answers
77 views

Why should I take this Contour for $I = \int_0^{\infty} \frac{dx}{1+x^3} $? (Analytic Continuation)

When discussing analytic continuation, my lecturer used the following example, $$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$ I have in my notes that the contour was taken as below. I must admit I was ...
2
votes
0answers
33 views

Is this integral automatically zero?

If I integrate $\int e^{iz}\,dz$ for z complex, along the positive real line, then is the imaginary part of the integral $i\int \sin(x)\,dx$ automatically equal to zero (integration only along the ...
1
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0answers
20 views

Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form \begin{equation} F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} \end{equation} where, $a$ and $b$ are known constants, $m$, $n$, ...
3
votes
2answers
104 views

How do I evaluate the following integral $\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m}$?

I am interested in the following integral $$\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m},$$ given that $m>n/2$ (this is just what I wrote so that the integral converges. If this is not ...
2
votes
1answer
61 views

Use contour integration to compute the Fourier transform,

The problem statement is: Use contour integration to determine the Fourier transform, $\large \hat f(ξ)=∫_{-\infty}^{\infty}f(x)e^{−iξx}dx$, of $\large f(x)=\frac{1}{2−2x−x^2}$. Some issues that I ...
0
votes
0answers
27 views

In P.V. contour integration in complex analysis, using a wedge vs. a keyhole contour

When is it clearly better -- perhaps even necessary -- to use a keyhole contour, instead of a wedge contour? The wedge contour minimizes computation of residues, as we can choose it so that it ...
0
votes
0answers
29 views

show that $\int_{C_R} \frac{z e^{iz}}{1+z^2}dz$ tends to zero

Show that $\int_{C_R} \frac{z e^{iz}}{1+z^2}dz$ where $C_R$ is the half circle in the upper half plane with radius $R$ tends to $0$ as $R$ goes to infinity. My professor showed me something with ...
0
votes
2answers
66 views

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem Ok so I have done a couple of these but I'm stuck on this one. I want to use $$ \int_0^\infty \frac{ze^{iz}}{1+z^2}dz $$ ...
0
votes
0answers
21 views

On the right half-plane, what is an upper bound for $\frac{1}{\log(z+2)}$?

I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for $\frac {1}{log(z+2)}$ on the semicircle in the right half plane is just $\frac ...
2
votes
2answers
55 views

Contour integral of $\log(z)/(z+a)^2$ around z=0

My question is primarily conceptual: Consider a function $f(z)$ which has a branch cut from $z=0$ to $z=\infty$ along the positive Re(z) axis. If I wish to integrate it along a small, clockwise circle ...
2
votes
2answers
112 views

Complex integration using singularities

I'm working on Ablowitz and Fokas' Complex Variables. On section 3.5 on singularities, problem 2 reads: Evaluate the integral of f(z) over the unit circle centered at the origin: a) $f(z)=z/(z^2 + ...
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0answers
20 views

The use of Cauchy Goursat Theorem for a contour

I have been attempting this question for some time. My reason is as follow: The contour C (−i, −2 − i, −2 −4i, −4i) is a rectangle on the left of the Imaginary axis (vertical axis). For any z ...
1
vote
1answer
45 views

Is a binomial expansion of the denominator valid here?

I am trying to prove this integral tends towards zero (a variation of the Hankel contour problem). I am not sure if my approximation is valid here, My integral round a circular pole at the origin is, ...
2
votes
2answers
86 views

Evaluating the improper integral $\int_{0}^{\infty} \frac{x^3}{e^{x}-1} dx$ [duplicate]

I read somewhere that $$ \int_{0}^{\infty} \frac{x^3}{e^{x} - 1} = \frac{\pi^4}{15}$$ Does anyone see a way to prove this? My first idea was doing a contour integration and use the residue theorem, ...
1
vote
2answers
76 views

Cauchy Integral Formula (When f(z) is not analytic everywhere inside C)

enter image description here For the question in the photo, I have thought about it for a long time. My idea is as follow: C (|z| = 2) is a positively oriented simple connected contour. The point ...
3
votes
1answer
50 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
4
votes
1answer
84 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
0
votes
1answer
43 views

How to Integrate$\int_C\frac{Log~z}{(z-i)^2}$ where $C:|z-i|=1$?

Let $C:|z-i|=\alpha$,where $0<\alpha<1$ I need to evaluate this expression $~\int_C\frac{Log~z}{(z-i)^2}$ what I tried to do was representing $Log~z$ as a Laurent series on ...
7
votes
3answers
151 views

How to evaluate $ \int_0^\infty \frac{\log x}{(x^2+a^2)^2} dx $

Evaluate $$ \int_0^\infty \frac{\log x}{(x^2+a^2)^2} dx $$ $$(a>0) $$ How can I use contour appropriately? What is the meaning of this integral? (additionally posted) I tried to solve this ...
1
vote
1answer
40 views

A question about the derivation of the Residue Theorem in Complex Variables Demystified

My question is about the last equality in the following derivation: f(z) is assumed to be a Laurent expansion about a singularity at $z_0$. $\oint f(z)dz= \oint \sum_{n=-\infty}^{n=\infty} ...
1
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0answers
54 views

Name of multidimensional propagator integral

$$ I_{n,m}(\boldsymbol{x},\boldsymbol{\tau}) = \dfrac{1}{(2\pi)^{n+m}} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \dfrac{e^{i(\boldsymbol{p}\cdot \boldsymbol{x}-\boldsymbol{q}\cdot ...
1
vote
1answer
60 views

How to compute this contour integral with a modulus sign in the integrand,

Evaluate the integral $$∫_{∣z∣=ρ} \frac {1}{|z−a|^{2}}|dz|$$ where ρ is a positive number, a is a complex number, and |a|<ρ. I welcome any hints on how to get started on this problem. The ...
2
votes
2answers
93 views

Contour integration of log over polynomial with fractional power

I've stumbled upon one integral which is rather challenging because of the fractional power of t: \begin{equation}\int_0^\infty \log(1+tx)t^{-p-1}dt, \end{equation} where $p\in(0,1)$ and $x>0$. ...
1
vote
3answers
179 views

Example using Cauchy's Theorem and integrating over a contour

This is an example out of Stein and Shakarachi Complex Analysis. My question is, Why do they choose $f(z) = \frac{(1 - e^{iz})}{z^2}?$
3
votes
0answers
42 views

Closed form of integral using contour integration

Here is the integral I am interested in evaluating using contour integration: Prove that: $$\int_0^\infty \frac{{\rm d}x}{(1+x^2)(1+x^r)}=\frac{\pi}{4}$$ That is that the above integral is ...
0
votes
1answer
58 views

Finding a contour to evaluate$\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+a^2}\,dx$

I am looking to evaluate to evaluate the real integral $I=\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+a^2}\,dx$ ($a>0$) using Cauchy's residue formula. My strategy is to use the residue theorem to ...
3
votes
0answers
58 views

Integral Representation for the Fox-H function on several variables

I have a problem that involves the H-function of several variables, and I have noticed that the implementation of such function when the number of variables are relatively high (greater than 5) must ...
0
votes
1answer
33 views

Evaluate the complex contour integral

Evaluate the complex contour integral $$\int \limits_C Log(z) dz$$ where $Log(z)$ denotes the principal complex logarithm and $C$ is the straight line from $z=1 +i$ to $z= 3+ 3i$ My attempt: We ...
-3
votes
1answer
52 views

Pick the $\zeta(3)$ contribution from Gamma function countour integral

I edited the post and title. How do we see that given $$ Z= \oint \frac{d \epsilon}{2\pi i} (z\bar z)^{-\epsilon} \frac{\pi^4 \sin 5\pi \epsilon}{\sin^5 \pi \epsilon} \left|\sum_{k=0}^\infty (-z)^k ...
4
votes
2answers
119 views

Inverse Laplace transform with branch cut

For the purpose of my research on persistent random walks I need to compute the inverse Laplace transform of $$ F(s)=\frac{\mathrm e^{-b\sqrt{s^2-1}}}{s^2-1}.$$ I looked up in tables of integral ...
0
votes
1answer
43 views

Laurent series expansion of $f(z)=\frac{1}{1-2^{1-z}}$ in the annulus $D_r=\{\ z\in\mathbb C\ |\ r<|z-z_0|<R\ \}$

How do you compute the Laurent Series of $f(z)=\frac{1}{1-2^{1-z}}$ in the neighborhood of the point $z_0=1$ converging inside the annulus $D_r=\{\ z\in\mathbb C\ |\ r<|z-z_0|<R\ \}$? ...
0
votes
2answers
68 views

Determine when the arc-length is (in)finite.

Problem: Let $\phi : [0,1] \to \mathbb C$ be given by $$\phi(t) := t + it\sin\left(\frac{\pi}{t^p}\right)$$ for $t \in (0,1]$, and define $\phi(0) := 0$. Then $\phi$ is continuous. For what values ...
4
votes
0answers
175 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
0
votes
1answer
71 views

Cauchy's theorem and a contour integral

$$\frac1{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} ds \, \frac{e^{-\sqrt{a s}}}{c s+s^{3/2}} \cos{\sqrt{b s}} \, e^{s t}$$ where $a, b$ and $c$ constant. To evaluate this, I used Cauchy's ...
3
votes
0answers
52 views

How to evaluate integral $\int_{-\infty}^{\infty} dx x e^{iax}e^{-ic\sqrt{(x^2 + b^2)}}$?

Like the title says, I am trying to evaluate $\int_{-\infty}^{\infty} dx \cdot x e^{iax}e^{-ic\sqrt{(x^2 + b^2)}}$ I'm not a mathematician by trade, but I ran across this integral in my research and ...
3
votes
1answer
61 views

How do I solve this complex integration problem?.

I want to find the value of $$I=\int_{|z|=r}\frac{|dz|}{|z-z_0|^4},$$ where $|z_0|\neq r>0$.
2
votes
1answer
25 views

Basic question regarding change of variable in contour integral

I am looking at the following integral $$ \int |x^s| ds, $$ where the integral goes from $\sigma -iT$ to $\sigma + iT$ (along $Re(s) = \sigma$). I am expecting that the integral is real because the ...
1
vote
1answer
40 views

Basic question regarding gamma function and contour integration

Here is something basic from complex analysis, and I was wondering if someone could possibly explain me how the following worked. Gamma function is defined for $Re(x)>0$ that $$ \Gamma(x) = ...
1
vote
1answer
15 views

Line integral of conservative field in polar coordinates

I am solving the vector equation: $$\vec \nabla P(r,\phi) = \vec f(r,\phi)$$ where $\vec f$ is conservative, in polar coordinates. Am I allowed to the following? $$\partial_r P= f_r$$ ...
1
vote
0answers
39 views

Difficult integral in terms of generalized functions

I am struggling to evaluate the following integral \begin{align} I = \int_{-\infty}^{\infty} \frac{e^{i\kappa_0\sqrt{r^2+t^2}}}{(r^2+t^{2})^{3/4}}e^{-ik t} d t. \end{align} I have tried standard ...
0
votes
0answers
52 views

Multivariate Complex Gaussian Integral

Consider the integral $$ \int_{\mathbb{R}^n}dx\,e^{-\frac12 x^TAx}=\frac{(2\pi)^{n/2}}{\sqrt{\det A}} $$ where $A$ is a symmetric $n\times n$ complex matrix with positive definite real part. ...
0
votes
0answers
59 views

Evaluating complex integral

I want to evaluate $$I = \int_0^\infty \frac{\ln z ~dz}{z \sqrt {1+z}}.$$ So I consider integral $$I^* = \int_0^\infty \frac{\ln ^2 z ~dz}{z \sqrt {1+z}}$$ in the contour: By the residue theorem ...