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

learn more… | top users | synonyms

3
votes
0answers
100 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
1answer
141 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
3
votes
1answer
157 views

Choice of branches for contour integration.

Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of $f(z)$ are $z\leq 0$ (presumably for the logarithmic term), and ...
2
votes
1answer
135 views

Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C} $ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
0
votes
1answer
43 views

Finding the types of singularities of $\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$

I want to find the types of singularities of $$\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$$ the point is $z=1$ I know that: $$f(z)=\frac{p(z)}{q(z)},q(a)=0,p(a)\neq 0,p(z)$$ so $p(z)$ analytic in $a$ ...
8
votes
3answers
284 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
11
votes
2answers
273 views

Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
3
votes
2answers
136 views

Does $\lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0$?

Let $C_{R}$ be the upper half of the circle $|z|= R$. Does $ \displaystyle \lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0 $? Jordan's lemma is not applicable here. And I'm not sure how to get a ...
4
votes
0answers
97 views

Determining “good” contours for evaluating integrals

This is more of a general question, but I'll lead with an example. Suppose we wish to evaluate $\displaystyle \int_0^{\infty} \dfrac{1}{1+x^7} dx$ The goal, it seems, is to find nice contours which ...
2
votes
0answers
118 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
1answer
123 views

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the ...
12
votes
3answers
404 views

Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$

Evaluate the following integral $$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$ This function does not have an elementary anti-derivative. How can we solve this?
2
votes
2answers
218 views

How to find $\int_{0}^{1}\frac{1}{x^{2}+2x+2}\mathrm dx$ with contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}\mathrm dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some ...
2
votes
0answers
215 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 ...
0
votes
2answers
124 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
3
votes
2answers
160 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
2
votes
3answers
70 views

Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
6
votes
3answers
106 views

calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
1
vote
2answers
146 views

Contour integration: $\int_0^\infty \frac{\cos x}{\sqrt{x}}dx$

A problem from a complex analysis qualifier: Find $$\int_0^\infty \frac{\cos x}{\sqrt{x}}dx$$ My answer so far: We want to integrate the function $$f(z) = \frac{\cos z}{\sqrt{z}}dx = ...
0
votes
2answers
56 views

find general solution to the Differential equation

Find the general solution to the differential equation \begin{equation} \frac{dy}{dx}= 3x^2 y^2 - y^2 \end{equation} I get \begin{equation} y=6xy^2 + 6x^2 y\frac{dy}{dx} - 2y\frac{dy}{dx} ...
1
vote
1answer
157 views

Contour Integration (Choice of Contour)

Let $ \alpha \le 0 $ and $\sigma > 0$ . I want to choose a contour, including $ [\sigma - iR, \sigma+iR] $ , such that i can apply Cauchy's Residue theorem and evaluate: $$ \lim_{R \rightarrow ...
2
votes
2answers
158 views

Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$

I am tasked to evaluate the sum $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$$ Using contour integration. This is what I've done so far. Let $C_{N}$ be the square defined by the lines ...
2
votes
2answers
54 views

Problem with Mellin Barnes type integral

Using the Mellin Barnes technique for a certain Feynman integral, I arrive at $$ I= \frac1{2\pi i} \int\limits_{-i\infty}^{i\infty} dz\; \Gamma^4\left(\frac12 + z\right) ...
2
votes
1answer
102 views

Explicit contour integration gone wrong.

Consider the function $f(z):\mathbb{C}\to\mathbb{C}$: $$f(z)=\frac{4z}{1+z^2}$$ There are a few properties evident: The anti-derivative (with integration constant $c=0$) is given by: ...
0
votes
2answers
34 views

Constructing an antiderivative of a function if the contour integral depends on initial and final point

I am working on the following problem: Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
8
votes
3answers
305 views

Evaluate the integral $I=\int_{0}^{\infty}\frac{\ln^3{x}}{(1+x^2)(1+x)^2}dx$

Find this integral $$I=\int_{0}^{\infty}\dfrac{\ln^3{x}}{(1+x^2)(1+x)^2}dx$$ My try: let $x=\tan{t}$ then $$I=\int_{0}^{\frac{\pi}{2}}\dfrac{\ln^3{\tan{t}}}{(1+\tan{t})^2}dt$$ I am unable to simplify ...
1
vote
2answers
69 views

Does $ \int x^{-2} \, \mathrm{d}{x} $ have a singularity?

How do you integrate $ \dfrac{1}{x^{2}} $ from $ 0 $ to, say, $ a $? Can you get a principal value? What is the divergence: $ + \infty $ or $ - \infty $?
0
votes
1answer
47 views

$\int_C \frac{1}{z^2}\cot \frac{1}{z}\,dz = 2\pi i$

Let $C$ be the unit circle $|z|=1$ oriented counterclockwise. Wolfram Alpha seems to suggest that $$\int_C \frac{1}{z^2}\cot \frac{1}{z}\,dz = 2\pi i$$ Since 0 is not an isolated singularity of the ...
3
votes
0answers
85 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( {{ ...
0
votes
1answer
41 views

How to solve $\int_{S_3^+(0)} \frac{e^w+z}{z+2} dw$

In my lecture notes the following integral was computed: \begin{align*} \int_{S_3^+(0)} \frac{e^w+z}{z+2} dw. \end{align*} There is written: In order to use the Cauchy Integral formula, which is ...
2
votes
2answers
185 views

Residues to solve an improper integral

I'm asked to solve the following improper integral: $$\int_0^\infty \frac{\rm {Log}^2(t)}{1+t^2}dt. $$ Do I consider the function $f(z) = \frac{\rm{Log}^2(z)}{1+z^2}$ or some variant? Is the ...
3
votes
1answer
101 views

How to prove $\operatorname{Log}(z) = \log(|z|)+i\arg(z)$.

The value of the principal branch of the logarithm can be evaluated by the formula \begin{align*} \operatorname{Log}(z) = \log(|z|)+i\arg(z), \end{align*} where $\arg(z) \in (-\pi,\pi)$ and ...
5
votes
1answer
140 views

Prove $\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)} \operatorname dx$

Derive the integral representation $$\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)}dx$$ for $|a|\le \pi/2$.
1
vote
1answer
101 views

Evaluating $I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$ for real $a$ and real $n > 0$ with Jordan's Lemma

Given that $e^{-nx^2}$ does not have any singularities, I believe $I(a,n) = 0$. Is this a correct application of Jordan's Lemma? $$I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$$
3
votes
4answers
199 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
2
votes
1answer
255 views

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...
0
votes
1answer
48 views

Prove that $F_a(z) - F_b(z) = F_a(b)$

We had the following statement. Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
1
vote
0answers
24 views

How to show $\int_{\sum_{i = 1}^n \gamma_i} f = \sum_{i = 1}^n \int_{\gamma_i} f$

Let $G \subset \mathbb C$ be an open set, $f: G \to \mathbb C$ a continuous function and $\gamma: [\alpha, \beta] \to G$ a contour. Define the contour integral of $f$ along $\gamma$ to be ...
1
vote
1answer
65 views

Computation of a certain integral involving cyclotomics

How would one compute $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz$ in terms of k and n. If this is not possible, how would someone find a good approximation for this. ...
1
vote
5answers
119 views

Indented Path Integration

The goal is to show that $$\int_0^\infty \frac{x^{1/3}\log(x)}{x^2 + 1}dx = \frac{\pi^2}{6}$$ and that $$\int_0^\infty \frac{x^{1/3}}{x^2 + 1}dx = \frac{\pi}{\sqrt{3}}.$$ So, we start with the ...
1
vote
1answer
65 views

Contour integration in the complex plane gone wrong

Considering a function of complex variable $z$: $$f(z)=\frac{e^z}{z}$$ and a contour integral: $$\oint_C dz f(z)$$ such that the contour $C$ encircles the origin counterclockwise, it is clear from the ...
1
vote
1answer
280 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
1
vote
0answers
81 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
4
votes
2answers
82 views

Establish $\int_0^{\infty} \frac{x^a}{x^2 + b^2}dx = \frac{\pi b^{a-1}}{2 \cos(\pi a /2)}$ when $-1 < a < 1$

My attempt at a solution: (this is homework, btw) Let $f(z) = \frac{z^a}{z^2 + b^2}dz$ then the singularities of $f$ occur at $\pm ib$. $$ Res(f; ib) = \frac{z^a}{z + ib} \biggr |_{ib} = ...
3
votes
2answers
134 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
3
votes
1answer
226 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
0
votes
0answers
30 views

Argue away the contribution from the contour integral

$$J(\lambda)=\int\frac{exp(i\lambda z^2)}{z-2-i}dz$$ Consider the contour integral above, consisting of a straight line C1 at $y=0$ between $1<x<r$, C2 given by $x=r$ between $0<y<r$ and ...
1
vote
2answers
181 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
0
votes
5answers
103 views

Complex Integration Problem. Please help.

Please help me with this one. Calculate the integral: $$\int_0^{2\pi} \frac{\mathrm{d}t}{a\cos t+b\sin t+c}$$ as $\sqrt{a^2+b^2}=1<c$. I'm working on it for quite a while and somehow I can't ...
2
votes
2answers
221 views

Complex Line Integral along the unit circle

Show that: $$\oint_\gamma\frac{1}{z}\left(z+\frac{1}{z}\right)^{2n}\mathrm{d}z=2\pi i\cdot\binom{2n}{n},\quad\text{while > }\gamma=\{z\in\mathbb{C}\,|\,|z|=1\}\,\,(\text{unit circle})$$ So ...