All Questions
Tagged with complex-analysis improper-integrals
559
questions
4
votes
2
answers
293
views
Is there a way of finding a closed-form expression for $\int_0^\infty\frac{k^2J_0(k)^2\,\mathrm{d}k}{k^4+\left( k^2+x^2 \right)^2}$, $x\in\mathbb{R}$?
I am trying to find a closed-form expression for the following improper integral
$$
\int_0^\infty \frac{k^2 J_0(k)^2 \, \mathrm{d}k}{k^4 + \left( k^2 + x^2 \right)^2} \, ,
$$
where $x \in \mathbb{R}$....
1
vote
4
answers
130
views
Convergence of $\int_{0}^{\infty} \frac{\log\left(x\right)}{x^{2} - 1}{\rm d}x$
I'm trying to solve the following complex analysis problem:
$$
\mbox{Show that}\ \forall\ n > 1\mbox{, the integral}\quad \int_{0}^{\infty}\frac{\log\left(x\...
1
vote
1
answer
95
views
Fourier transform of incomplete gamma function
Ultimately I am interested in the Fourier transform of
$$
e^{-i\zeta}(-i\zeta)^{-2\epsilon}\Gamma(2\epsilon,-i\zeta)
$$
in a series expansion around $\epsilon=0$, so to first order in
$$
\lim_{\...
- 133
2
votes
6
answers
228
views
Show that $\int_{-\infty}^{\infty} \frac{x^2}{\left(x^2+a^2\right)^2} dx = \frac{\pi}{2a}$
I am trying to show that
$$\int_{-\infty}^\infty \frac{x^2}{\left(x^2 + a^2\right)^2} dx = \frac\pi{2a}$$
for $a > 0$ using the Residue Theorem.
The formula I am using says
$$\int_{-\infty}^\infty ...
- 3,187
3
votes
1
answer
122
views
Solve $\int_{-\infty}^{+\infty}\frac{1}{\cosh x}\ dx$ using residue theory [ANSWERED]
I was trying to solve this exercises which asked to first solve
$$I=\lim_{R\to +\infty}\oint_{\Gamma_R}\frac{1}{\cosh z}\ dz $$
where $\Gamma_R=\partial\{z=x+iy\in\mathbb{C}:-R\le x\le R, \ 0\le y\le \...
- 113
0
votes
0
answers
45
views
Trying to solve $I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx$
While trying to find an answer to this problem on the forum, I came across this integral:
$$ I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$
Where $c$ and $k$ are real numbers. I ...
0
votes
0
answers
95
views
complex analysis - Help with integrating $\int_0^{\infty} \frac{(\log x)^4}{x^2 + 1} \operatorname d\!x$ [duplicate]
I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane):
I have split the contour into 4 parts - the large semi-circle, ...
- 1
1
vote
1
answer
55
views
Contour integral of $\int_{γ} \frac{1}{|z-z_{0}|^\alpha} dz$ where $0<\alpha<1$ [closed]
I am studying Cauchy integrals and their value in points on the integration curve. The principal Cauchy value is defined on these points, however in part of the proof it is used that
$\int_{γ} \frac{1}...
1
vote
0
answers
33
views
Complex integral from quantum mechanics [duplicate]
I got this integral in one quantum mechanics problem: $$\int_{-\infty}^{\infty} e^{-i\cdot p_y y/\hbar}e^{-\beta y^2}dy$$ and the solution was apparently $\sqrt{\frac{\pi}{\beta}} e^{-\frac{p_y^2}{4\...
- 87
2
votes
1
answer
125
views
Integrating a Modified Bessel function of the second kind with a singularity
Does someone know how to handle the integral
$$\int_{-\infty}^{\infty} \frac{K_0\!\left(\lvert \tau \rvert \sqrt{q^2 \alpha ^2}\right)}{q^2}\cos (q x)\,\mathrm{d}q $$
$\alpha$ is a real number and $\...
2
votes
1
answer
341
views
On the 'reverse theorem' for functional equation of the Riemann zeta function
The following is the first part of Section 2.13. of Titchmarsh's book The Theory of the Riemann Zeta-Function:
My first question (orange-underlined) is: How $R_{\nu} = x^{-\frac12 s_{\nu}} Q_{\nu}(\...
- 281
4
votes
1
answer
183
views
Show that $\dfrac{1}{2 \pi i} \int_{c- i \infty}^{c+ i \infty} F(s) G(w-s) ds = \int_0^{\infty} f(x) g(x) x^{w-1} dx.$
Eq. 2.15.10 in Titchmarsh's book The Theory of the Riemann Zeta-Function comes with no proof:
Let f(x) and $F(s)$ be ralated by $$F(s) = \int_0^{\infty} f(x) x^{s-1} dx, \ \ \ f(x) = \dfrac{1}{2 \pi i}...
- 281
2
votes
2
answers
166
views
Product of Laplace transforms with disjoint Region Of Convergence (ROC)
Let the Laplace transform of $x(t)$ be $$X(s) = \int_{-\infty}^{+\infty}x(t)\exp(-st)dt, \ \ \ \ \ s\in\mathbb{C}$$This integral converges when $s \in \text{ROC}_1$. Also suppose that the Laplace ...
- 4,359
2
votes
2
answers
157
views
Choice of contour for Residue theory
As an application of Cauchy's Residue Theorem, integrals of the form $\displaystyle \int_{-\infty}^{\infty} f(x) ~d x$, when the bounds are $0$ to $\infty$ and the integrand has finite number of ...
- 379
5
votes
1
answer
181
views
How to evaluate $ \displaystyle \int_0^\infty \frac {\sin x}{1+x^3}dx. $?
I was wondering if we can use complex contour integration to evaluate the integral
$$
\int_0^\infty \frac {\sin x}{1+x^3}dx.
$$
Since the integrand is not even, we cannot extend the integration domain ...
- 2,015