For questions about integration methods that use results from complex analysis and their applications

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23
votes
2answers
588 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
21
votes
2answers
1k views

Summation using residues

In reference to this question about showing that the following interesting series takes on the value $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}$$ I tried ...
16
votes
7answers
2k views

Integration by means of complex analysis

Dear all: this time I have the integral $$\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$$and we must try to solve it using complex integration, residues, Cauchy's Theorem and the whole lot. (BTW, does ...
12
votes
1answer
198 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 $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
10
votes
5answers
605 views

Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$.

How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
10
votes
4answers
415 views

contour integration of logarithm

I must compute the following integral $$\displaystyle\int_{0}^{+\infty}\frac{\log x}{1+x^3}dx$$ Can someone suggest me the right circuit in the complex plane over which to do the integration? I ...
10
votes
1answer
135 views

Analytic function $f$ in $\overline{\mathbb{D}}$ satisfying $\left\lvert\,f'(\tfrac{1}{2})\,\right \rvert\leq 8.$

Let $f$ be an analytic function on the closed unit disk $\overline{\mathbb{D}}$. On its boundary $\partial \mathbb{D}$ it holds that $\lvert\,f(z) -z\rvert < \lvert z\rvert$. I now have to show ...
8
votes
2answers
538 views

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
8
votes
2answers
114 views

closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

$$I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$$ for $n=1$ I tried to use $\arctan x=u$ and by notice that $$\frac{1+\tan u}{1-\tan u}=\cot\left ( ...
8
votes
3answers
138 views

Cauchy's Theorem - Prove that $\sum_{n=1}^\infty \frac{1}{\lambda_{n}^2} $ = $\frac{1}{10}$

I seek to prove that $$\sum_{n=1}^\infty \frac{1}{\lambda_{n}^2} = \frac{1} {10},$$ by applying Cauchy Theorem to $$ f(z) = \left(\frac{z\tan(z)}{z-\tan(z)}+\frac{3}{z}\right) \frac{1}{z^2},$$ ...
7
votes
1answer
746 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
7
votes
2answers
109 views

Proof using Cauchy Integral Theorem

Suppose that I have the following integral: $$ \int_L \frac {dz}{z^2+1} $$ I need to show that this is equal to $0$ if $L$ is any closed rectifiable simple curve in the outside of the closed unit ...
7
votes
2answers
217 views

Choosing parametrization for complex integration with two branch cuts

I am particularly interested in how Ron Gordon came up with the parametrization in his anser to this question: Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( ...
7
votes
1answer
850 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
6
votes
3answers
466 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$ Any help will be much appreciated.
6
votes
1answer
82 views

Determine the Integral $\int_{-\infty}^{\infty} e^{-x^2} \cos(2bx)dx$

Please do not mark this question as a duplicate. I have to solve this with a different method that I don't believe has been discussed about this particular question (at least to my knowledge). I am ...
6
votes
4answers
294 views

Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
6
votes
2answers
129 views

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in ...
6
votes
2answers
85 views

Evaluate $\int_{|C|=2} \frac{dz}{z^2 + 2z + 2}$ using Cauchy-Goursat

I've split the integral around $z_1 = 1 - i$ and $z_2 = 1+ i$ using the contours $C_1$ and $C_2$: $ \int_{|C|=2} g(z) dz = \int_{C_1} g(z) dz + \int_{C_2} g(z) dz$ In this case, $g(z)$ for $C_1$ ...
6
votes
3answers
171 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
6
votes
3answers
87 views

If $f$ is entire and $\lim_{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant

I'm learning about complex analysis and need to verify my work to this problem since my textbook does not provide any solution: If $f$ is entire and $\lim_{z\to\infty} \frac{f(z)}{z} = 0$ show ...
6
votes
1answer
169 views

Global Residue Theorem in $\mathbb{CP}^2$.

Consider the following meromorphic form defined on $\mathbb{CP}^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
6
votes
3answers
108 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
6
votes
1answer
75 views

Evaluating an indefinite integral using complex analysis

Using tools from complex analysis, I have to prove that $$ \int_0^{\infty} \frac{\ln x}{(x^2 + 1)^2}\,dx = - \frac{\pi}{4}.$$ But I'm not really sure where I should start. Any help would be ...
6
votes
1answer
160 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
6
votes
0answers
217 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
5
votes
2answers
456 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 ...
5
votes
3answers
516 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
5
votes
3answers
102 views

There is an entire function $g$ such that $f(z)=g\left(z^{n}\right)$.

Let $f$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f\left(\xi z\right)=f(z)$ for all $z\in \mathbb{C}$. Show that there is a entire function $g$ ...
5
votes
2answers
318 views

A (basic?) contour integration problem

I am trying to prove the following using complex analysis: $$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$ I am told to use the following function: ...
5
votes
1answer
73 views

show that $\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$

$$\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$$ using real or complexe analysis where $B_{2u}$ is bernoulli number
5
votes
1answer
258 views

Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
5
votes
4answers
770 views

What is the value of $\int_{\gamma} \bar{z} dz$?

I could use some help in calculating $$\int_{\gamma} \bar{z} \; dz,$$ where $\gamma$ may or may not be a closed curve. Of course, if $\gamma$ is known then this process can be done quite directly (eg. ...
5
votes
2answers
162 views

“Convergent” Integral in Davenport's Multiplicative Number Theory

I am currently learning analytic number theory using Davenport's Multiplicative Number Theory book, and at some point I believe something silly is happening. I have great faith that I am wrong AND ...
5
votes
2answers
515 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
5
votes
2answers
228 views

Why do we need a branch cut for $\int_0^{\infty} \frac{x^{\frac{1}{2}}}{{(1 + x)^2}}dx$?

What is the significance of the $x^{\frac{1}{2}}$ in the numerator of this integral. I have read this kind of integral requires taking a branch cut. Why do we need a branch cut, what does it enable us ...
5
votes
2answers
52 views

Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

I need to evaluate the integral $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)} dz$ as a function of $R>0$. I may omit values of $R$ for which the denominator turns to $0$. Now, ...
5
votes
2answers
83 views

Computing alternating sum using contour integration

By considering the integral of: $$\left(\frac{\sin\alpha z}{\alpha z}\right)^2 \frac{\pi}{\sin \pi z},\quad \alpha<\frac{\pi}{2}$$ around a circle of large radius, prove that: ...
5
votes
3answers
102 views

Prove that $\oint_{|z|=r} {dz \over P(z)} = 0$

I got stuck on this problem, hope anyone can give me some hints to go on solving this: P is a polynomial with degree greater than 1 and all the roots of $P$ in complex plane are in the disk B: ...
5
votes
1answer
58 views

Use Cauchy's Theorem to show that if $\int_{0}^{\infty}f(x)dx$ exists, then so does $\int_{L}f(z)dz$

Suppose that $f(z)$ is analytic at every point of the closed domain $0 \leq arg z \leq \alpha$ $(0 \leq \alpha \leq 2 \pi)$, and that $\lim_{z \to \infty}z f(z) = 0$. I need to prove that if the ...
5
votes
1answer
159 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
5
votes
1answer
578 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
5
votes
1answer
75 views

Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and ...
5
votes
2answers
202 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
5
votes
2answers
108 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
5
votes
1answer
133 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
4
votes
2answers
208 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
4
votes
1answer
167 views

What do Fourier Transforms actually do?

I will illustrate this with a simple example: Consider the exponential decay function $$f(t)=\begin{cases} 0 & \ t\lt 0 , \\ A e^{-\lambda t} & \ t\ge 0 \end{cases}$$ Where $\lambda ...
4
votes
1answer
54 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
4
votes
2answers
1k views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...