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

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0
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2answers
122 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
157 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
100 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
144 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
55 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
151 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
157 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
51 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
97 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 $?
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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
168 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
97 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
137 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
100 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
198 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
251 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
117 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
62 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
267 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
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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
81 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
132 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
213 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
177 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
202 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 ...
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1answer
68 views

Please help me with this Integral

Calculate the integral (complex): $$\oint_{D(0,1)}\overline ze^z \mathrm dz$$ While $D(0,1)$ is the unit circle.
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1answer
61 views

Work done by gravitational force

In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse ...
2
votes
2answers
225 views

What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
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2answers
50 views

Inverse Laplace of $\frac{s^3}{2+s^3}$

How I can find the Inverse Laplace of $\displaystyle \frac{s^3}{2+s^3}$ Thanks
4
votes
1answer
63 views

Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved. My ...
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0answers
52 views

Integration of rational function on Banach algebra

I do not follow the proof of this Theorem Theorem Suppose$R(\lambda) = P(\lambda) + \sum_{m,k}c_{m,k}(\lambda - \alpha_m)^{-k}$ is a rational function with poles at the points $\alpha_m$. ($P$ ...
17
votes
4answers
462 views

For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb ...
1
vote
1answer
139 views

Evaluating series by contour integration, the residue theorem, and cotangent

I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is ...
5
votes
3answers
180 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
1
vote
0answers
76 views

Contour Integrals Complex Analysis

Evaluate $$\int_C\!\frac{\cos(z)}{z(z+2)}dz$$ where $C$ is the square with side $4$ centered at $z=0$ oriented clockwise.
3
votes
1answer
125 views

Compute $\int_{|z|=1}\frac{\log z}{z}dz$.

Here is a question about contour integration in complex analysis: Compute $$\int_{|z|=1}\frac{\log z}{z}dz$$ I am not sure if I understand the question since the logarithm must be defined in a ...
0
votes
0answers
142 views

Difficult Fourier transform

While looking at non-local modifications to wave propagation in 2d I have run into the following integral $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega dk \ln(k^2-\omega^2)e^{-i\omega ...
3
votes
1answer
41 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
0
votes
1answer
52 views

Complex Analysis: Principle Part and Evaluating Integrals

I have two quick questions. Identify the pole in the following function and find the res of said function at it's pole. $(1)$\ $G(z)=\frac{\cos(z)}{\sin(z)}$ Here, ...