Questions tagged [complex-integration]
This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.
3,162
questions
2
votes
1
answer
78
views
Where is the error in evaluating this integration:$\int^{\infty}_{-\infty}\frac{\cos(2x)}{(1+x^2)^2}$
Where is the error in evaluating this integration:$$\int^{\infty}_{-\infty}\frac{\cos(2x)}{(1+x^2)^2}dx$$
we have:
$$\int_{C_R}\frac{\cos(2z)}{(1+z^2)^2}dz=\int_{-R}^{R}\frac{\cos(2x)}{(1+x^2)^2}dx+\...
- 2,350
0
votes
0
answers
66
views
Sign of a complex integral
If one consider the complex value function
$$
f(z)=\frac{1}{(z-1)^{3/2}(z-2)^{3/2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$.
Consider
$$
\oint f(z) dz,
$$
where the contour is taken to ...
- 1,139
1
vote
1
answer
106
views
Complex integral with fractional singularities
If one consider the complex value function
$$
f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why
$$
2\int_1^2 f(x)dx=\oint f(...
- 1,139
0
votes
1
answer
92
views
understanding the integral $\oint_{|z| = 1}z^{-1}dz$
This is a somewhat vague question.
I am aware that the integral in the title can be computed via a parametrization and that its value is $2\pi i$. The $i$ comes from differentiating $\exp(it)$ and the ...
- 387
4
votes
1
answer
135
views
How to prove $\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz$ in the sense of Borel summation?
As the title shows, I would like to prove this identity in the sense of Borel summation,
$$\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz,$$
providing ...
- 43
4
votes
1
answer
50
views
Directly parameterize to calculate two integrals
Directly parameterize to calculate the integrals:
(a) $\int_{K} \sin(\bar{z}) dz$, where $K$ is the line segment from the point $i$ to the point $3i$.
(b) $\int_{K} \sqrt{z} dz$, where $K = \{ z \in \...
- 49
1
vote
1
answer
102
views
Using the residue theorem to compute two integrals [closed]
Classify the singular points for the function under the integral and using the residue theorem, compute:
(a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$
(b) $$ \int_{|z|=2} \sin\left(\frac{...
- 45
1
vote
1
answer
85
views
Compute the real integral $ \int_{0}^{\infty} \frac{x \sin(2x)}{x^2 + 1} \, dx. $ [duplicate]
Using the integration of the function $f$ with the prescription $f(z) = \frac{ze^{2iz}}{z^2 + 1}$ along the positively oriented boundary of the region $D = \{ z \in \mathbb{C} \mid |z| \leq R, \...
- 45
3
votes
1
answer
92
views
Classify the singular points for the function under the integral and using the residue theorem
Classify the singular points for the function under the integral and using the residue theorem, calculate:
(a)
$\displaystyle\int_{|z|=3} \frac{1 - \cosh z}{z^6 + 2z^5} \, dz $ and
(b)
$\displaystyle \...
- 89
0
votes
2
answers
95
views
Deformation of path for complex integrals
I have been reading the complex analysis part from the book "Advanced Engineering Mathematics by Erwin Kreyszig" and have been confused regarding this theorem's application.
Independence of ...
4
votes
1
answer
159
views
Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.
Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.
(a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
- 85
1
vote
0
answers
63
views
$\int_K |z| dz$ and $\int_K \text{Im}(z) dz$ for different $K$-s
Calculate the integral:
(a) $\int_K |z| dz$, if $K$ is the line segment from the point $0$ to the point $2 - i$.
(b) $\int_K \text{Im}(z) dz$, if $K = \{ z \in \mathbb{C} \mid |z| = 1 \land \text{Im}...
- 45
1
vote
0
answers
40
views
Mellin transform of exponential and logarithm
I am trying to calculate the Mellin transform of the function $f(x) = e^{-ax}\ln\left(1+x\right)$. The mathematica gives me the answer
$$\int_{0}^{\infty}x^{s-1}f(x)dx = \frac{G_{3,5}^{5,2}\left(\frac{...
- 3,263
0
votes
1
answer
42
views
How to get the sign right for branch-cut contour integration of the standard free-field propagator
(Apologies for any awkwardness. This is my very first post.)
This is a question about how to get the sign right for the classic integral dealt with here
Keyhole Contour with Square Root Branch Cut on ...
- 1
0
votes
0
answers
10
views
Asimptotic contur integral from q-pochhummer symbol power -0,5 multiplied with exponent
I tried to get asimptotical behaivior of the function $f(z)$ when complex $ z \to \infty $ of the following expression:
$$ f(z) = (z; q)_{\infty}^{-1/2}, 0<q<1 $$
I would be very grateful for ...
