Questions on the evaluation of integrals along a locus in the complex plane.
7
votes
1answer
62 views
Help with understanding the evaluation of a real integral using complex analysis
This morning I decided to look up some complex analysis, and I came across this Wikipedia section, where the following integral is evaluated:
$$\int_0^3 \frac{x^{3/4}(3-x)^{1/4}}{5-x}dx$$
There are ...
6
votes
1answer
114 views
Is this a correct way to calculate $\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx?$
I have this integral to calculate: $$I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx.$$
I think I have done it, but I would like to make sure my solution is correct.
I take the function ...
1
vote
1answer
47 views
Contour integral of $\displaystyle\int_\gamma \dfrac{1}{(2z+1)(z+3)^2}$
Im a little confused by the following integral question
Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the anti-clockwise direction.
$\displaystyle\int_\gamma ...
1
vote
1answer
18 views
Contour Integrals for positively circular contour
Find the contour integral of $\frac{1}{(z^2+1)^2}$ for the positively oriented circular contour $|z-Ri|=R$, for every positive real number $R>\frac{1}{2}$.
I don't know how to set up the ...
0
votes
1answer
52 views
Justification in change of variables
it would be fantastic if anyone could help me with the following problem:
I have the integral
$$\operatorname{Im} \left( \int^\infty_0 e^{it} t^{s-1} \mathrm{d} t\right)$$
and I wish to make the ...
0
votes
1answer
44 views
Contour Integrals and positively oriented circles
If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{
\begin{array}{lr}
0 & n=\pm1, \pm2, ...\\
2\pi i & n=0\\
...
0
votes
1answer
68 views
Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane
Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ ...
0
votes
1answer
71 views
Evaluate $\int_{\gamma_R}f(z) dz$ when $R>2$
Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$
and $\gamma _R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and $0\leq t\leq\pi$ the straight line ...
4
votes
0answers
95 views
contour integration around a dogbone/dumbbell contour
I'm getting the correct answer, but I'm not confident in what I'm doing.
I want to evaluate $\displaystyle\int_{0}^{1} \frac{1}{\sqrt[3]{x^{2}-x^{3}}} \ dx $ using contour integration.
I'm going to ...
4
votes
0answers
78 views
Contour integration of $\int_{-\infty}^{\infty}e^{iax^2}dx$
Consider the following integral:
$$\int_{-\infty}^{\infty}e^{iax^2}dx$$
Here I believe we have to consider the two cases when $a<0$ and $a>0$, as they need different contours. For $a>0$ ...
4
votes
0answers
200 views
contour integral with singularity on the contour
I want to compute the following integral
$$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$
The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
3
votes
0answers
28 views
Contour integral representation of Confluent Hypergeometric Function
My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$:
From ...
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
2
votes
0answers
71 views
quick integration notation question
What exactly does the base of this integral sign refer to?
$$\int\limits_{[0\to 1]}$$
Is that an arbitrary contour from the point $0$ to the point $1$? (this is complex analysis)
ps. Here's the ...
2
votes
0answers
53 views
Why is contour integration defined only for continuous functions?
Why is contour integration defined only for continuous functions?
Ordinary Riemann integration has no such stipulation (for eg. the Thomae function is discontinuous yet integrable.)
2
votes
0answers
103 views
difficult integral involving arcsin(x)
I have a difficult integral to compute. I know the result by guessing the answer, but need to know the method of calculation. The integral is
$\int_{a}^{b} dp\, p \frac{2\sin^{-1} \big(\frac{1}{p} ...
1
vote
0answers
144 views
Inverse Laplace Transform by contour integration
In question 1) we get Laplace transform of $$ g(t) = t^a $$ is:
$$
\hat g(t)= {1/s^{a+1}}\int_0^\infty e^{-t}x^a
$$
then I was stuck at question 2) which asks me to evaluate the inverse laplace ...
1
vote
0answers
73 views
What is suitable contour shape for $\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$
$\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$ . What kind of contour is suitable for this integral?
1
vote
0answers
116 views
Integrating over a closed contour following vector field
Integrate over a closed contour $c$
$$\oint_c d\vec{r}\times\vec{a}, \quad \vec{a}=-yz\vec{i}+xz\vec{j}+xy\vec{k}$$
where $c$ is cross-section of following two surfaces
$$x^2+y^2+z^2=1$$
and
$$y=x^2$$ ...
1
vote
0answers
102 views
Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null
Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null.
I think it has something to do with the fact that $f'$ is bounded in any ...
1
vote
0answers
142 views
How to choose a proper contour for a contour integral?
When analyzing real integrals with contour integrals, how does one choose a proper contour integral?
Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
0
votes
0answers
13 views
Contour integration with branch cut
This is an exercise in a course on complex analysis I am taking:
Determine the function $f$ using complex contour integration:
$$\lim_{R\to\infty}\frac{1}{2\pi ...
0
votes
0answers
24 views
Definition: “A contour respects causality”
When doing a contour integral, what does "the contour respects causality" mean?
0
votes
0answers
76 views
About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$
Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
0
votes
0answers
62 views
About the inverse laplace transform of sinc function
How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ?
Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ .
Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
0
votes
0answers
70 views
Contour Integrals and counterclockwise
$\int_C (z-z_0)^{(n-1)}\ dz$ for any integer $n$, where $C$ is the contour once around the circle $|z-z_0|=1$ counterclockwise and $z_0$ is any point in the plane. Also give the values of the ...
0
votes
0answers
143 views
$\int e^{-x^2}dx$ with contour integration
Is it possible to prove that
$$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt\pi$$
integrating $e^{-z^2}$ along a suitable contour?
0
votes
0answers
97 views
Concept of Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...
