Tagged Questions
0
votes
1answer
32 views
Evaluate this integral without using the residue theorem
I want to evaluate this integral
$$ I = \int_c \tan z + \frac{\csc z}{z} dz $$
$$ c :|z| = 1 $$
apparently tanz is analytic in this region so its integral equals to zero
now
$$ I = ...
1
vote
0answers
79 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
0
votes
1answer
82 views
Calculating Residues
I want to calculate this integral
$$I:=\int dk^{0}\frac{e^{-ik^{0}(x^{0}-x'^{0})}}{\left(\left(k^{0}\right)^{2}-|\vec{k}|^{2}\right)} $$
for that I recall the Residue Theorem:
$$I=2\pi i \left\{ ...
1
vote
1answer
98 views
finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis
I am trying to find $\int_0^\infty \dfrac{dx}{1+x^4}$ by setting it equal to $\dfrac{1}{2}\oint_C \dfrac{dz}{1+z^4}$ and solving that. By a computer program I've calculated it to be $\approx 1.11072$; ...
2
votes
1answer
198 views
How to evaluate this complex integral !?
We have the following complex integral :
$$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$
Where $x\in\mathbb{R}:x>1$. i tried closing ...
2
votes
1answer
235 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 $\pi^3/8$, as you can verify (for example) introducing the function
$$
...
3
votes
1answer
188 views
Complex analysis integration with residues.
I have to show that $$\int_{0}^{2\pi}\frac{d\theta}{(a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta)^{2}} =\frac{ \pi(a^{2}+b^{2})}{a^{3}b^{3}}$$ where $a,b>0$. I have tried using double angle formulas ...
1
vote
0answers
161 views
Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0
The ultimate aim is to solve the following integral:
\begin{equation}
\label{eq:Icos1}
\begin{aligned}
I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
-1
votes
2answers
87 views
$\int_{-1}^{1}\exp(-at^2)/t^2 dt$ using residue theorem
How to calculate $$\int_{-1}^{1}\frac{e^{-at^2}}{t^2}dt$$ where $a>0$, using residue theorem?
