Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

Filter by
Sorted by
Tagged with
1 vote
0 answers
29 views

Contour Integral with square root

I'm a master degree theoretical physics student and while working on my thesis I've encountered the following guy: $$\int_0^{\infty}dx\frac{e^{-ax^2+ibx}}{\sqrt{x}}$$ with $a,b>0$. I wanted to ask ...
user avatar
3 votes
0 answers
76 views

Contour integration of $\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$

Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$ How to calculate it? I never worked with integrals of this type.
user avatar
  • 33
0 votes
0 answers
52 views

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; Re a > 0

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; $Re (a) > 0$ I found two questions where a > 0 but in my case I have the following condition: Re a > 0 (It ...
user avatar
  • 33
1 vote
0 answers
50 views

Three complicated integrals lead to an anodyne expression

Someone being forgotten by me has claimed: $$\int_{0}^{1} \frac{1}{(1+x^2)\sqrt{1-x^2}\sqrt{4-x^4}\sqrt{9-x^4} } \text{d} x -\int_{0}^{1} \frac{1}{(3+x^2)\sqrt{1-x^4} \sqrt{2+x^2}\sqrt{4+x^2}\sqrt{5+...
user avatar
1 vote
1 answer
31 views

How to derive the following estimate from example 3.9 in Bruce Palka's textbook: $|\int_\beta\frac{e^{iz}}{z}\,dz| \leq \frac{\pi(1-e^{-r})}{r}$

The following inequality occurs in example 3.9 on p. 336 of Bruce P. Palka's An Introduction to Complex Function Theory (Springer, 1990). Let $r\in(1,\infty)$, and define $\beta: [0,\pi]\rightarrow\...
user avatar
  • 10.4k
0 votes
0 answers
28 views

Does every pole have non-zero residue?

I am self-studying complex analysis.I am currently studying isolated singularity.There are $3$ kinds of singularities viz. removable singularity,pole and essential singularity.We know that if $z_0$ is ...
user avatar
0 votes
0 answers
18 views

Convergence conditions of the integral $\int_{-\infty}^{\infty} \frac{e^{cx}}{1+e^{x}} dx$.

I want to find the conditions such that $\int_{-\infty}^{\infty} \frac{e^{cx}}{1+e^{x}} dx$ converges. I am considering as contour the rectangle with vertices $\pm R$, $\pm R + 2\pi i$. \begin{align*}...
user avatar
0 votes
0 answers
17 views

Convert Bessel function contour integral definition to the imaginary line

I would like to solve the following integral $$I_{\nu}(x) = \frac{1}{{2\pi i}}\int_{-i\infty}^{i\infty} d\lambda \frac{e^{\frac{x}{2}(\lambda - 1/\lambda)}}{\lambda^{\nu+1}}.$$ The integrand is ...
user avatar
0 votes
1 answer
29 views

Evaluating $\int_0^\infty e^{-ax}\cos(bx)dx$ and $\int_0^\infty e^{-ax}\sin(bx)dx$ for $a>0$ using complex analysis.

I am a Mathematics student.Currently I am doing exercises of complex analysis from Stein Shakarchi's book.In the second chapter,there is a problem which is as follows: Evaluate $\int_0^\infty e^{-ax}\...
user avatar
0 votes
1 answer
23 views

When does the value of a complex contour integral depend on the choice of the contour of integration?

I have seen examples of complex contour integrals whose value depends on the choice of the contour of integration and some integrals where the value does not depend on the choice of the contour. Is ...
user avatar
2 votes
2 answers
43 views

General condition for the surface element be the same as the volume element, up to a dt

The surface element in spherical coordinates is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi$, and the volume element is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}r$. We see ...
user avatar
  • 1,621
2 votes
1 answer
32 views

Evaluating improper integrals with the help of contour integrals.

I am a graduate student.I have been studying complex analysis from Stein Shakarchi's book.In chapter $2$ ,there is an exercise which is as follows: As given in the hint,I assume that the countour is $...
user avatar
0 votes
1 answer
33 views

Should a compactly supported field have a Helmotz decomposition that is compactly supported?

Let $\bf F$ be a smooth vector field, which is null outside a finite compact domain $V$. By Helmoltz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} ...
user avatar
  • 1,621
1 vote
0 answers
31 views

Transforming every volume integral into a surface integral

Helmotz decomposition theorem says, on one hand, that every vector field $F$ sufficiently smooth can be decomposed into the sum of a solenoidal field $\nabla\times \bf A$ and a gradient field $\nabla \...
user avatar
  • 1,621
0 votes
1 answer
33 views

How to choose the correct integration path in complex integration? [duplicate]

I am doing some exercises in complex integration. I am given the two functions: $$\frac{\sqrt{x}}{x^2+4}$$ $$\frac{1}{\sqrt{x}(x^2+1)}$$ They must be integrated from $0$ to $+\infty$ with a complex ...
user avatar
  • 341
2 votes
0 answers
31 views

Easy way to compute repeated residue calculus?

I want to evaluate an integral, which can be in principle computed exactly. For $a>0$ and $k_1, k_2 >0$, define $G(x) = (x+ia)^3$. The integral that I have is $$I = \int_{\mathbb R^4} dy_1dy_2 ...
user avatar
  • 1,675
2 votes
0 answers
59 views

Integrate $\int_0^\infty e^{-\sqrt{x}} \mathrm{d} x$ using complex analysis

I'm trying to compute the integral $\int_0^\infty e^{-\sqrt{x}} \mathrm{d} x$ using complex analysis. I'm working on a problem for a complex analysis class that asks me to do so, but I'm struggling ...
user avatar
  • 7,909
5 votes
1 answer
136 views

Apparent complex integration paradox

I encountered the following integral : $$ \int_{\gamma} \frac{dz}{z-1-i}, $$ which has to be integrated along two straight-line contours : $\gamma_{1} : 2i$ to $3$ $\gamma_{2} : 3$ to $0$ and then ...
user avatar
1 vote
1 answer
60 views

Evaluate the following Contour Integral: $\int_C\frac{dz}{z^4+1}$ where $C$ is the circle $x^2+y^2=2x$

Question: Evaluate assuming the closed contour is traversed in the positive direction: $\int_C\frac{dz}{z^4+1}$ where $C$ is the circle $x^2+y^2=2x$ My Thoughts: Considering the circle $C:=x^2+y^2=2x$,...
user avatar
  • 2,390
0 votes
2 answers
64 views

Let $γ : [0, 1] → C$ be given by $ γ(t) := 2 + e^{2πit}$. Compute the path integral $\int_{\gamma} z^2$dz

$γ : [0, 1] → C$ be given by $ γ(t) := 2 + e^{2πit}$ Compute the path integral $\int_{\gamma} z^2 dz$ My solution: using the defintion of the path integral $\int_{\gamma} z^2 dz = \int^1_{0} (2+e^{...
user avatar
  • 479
0 votes
1 answer
76 views

Use residues to evaluate $\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx$

I need to use residues to evaluate $\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx$. So first, $$\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx = \int_{C_R} \frac{z^{1/3}}{1+z^2} + \int_{-R}^R \frac{x^{1/3}}{1+...
user avatar
3 votes
0 answers
89 views

Difference between $\int$ and $\oint$

If $C$ is a closed curve, what is the difference between $$\int_Cf(z)\; dz\text{ and } \oint_Cf(z)\; dz?$$ For example why on Wikipedia Cauchy's integral theorem is For a holomorphic function $f$ and ...
user avatar
  • 31
1 vote
1 answer
37 views

Evaluating $\oint_C{z^{-\frac{1}{2}}}e^{iz}dz$ to derive Fresnel Integrals.

I need to solve the integral $I=\int_{0}^{\infty}sin(u^2)du$(Fresnel Integral). I made some progress in this regard, but I had some doubts as to how to proceed from then on. Here is the approach i ...
user avatar
  • 99
0 votes
0 answers
49 views

Find $\int_0^\infty \frac{1}{1+x^5} \, dx$ [duplicate]

I need to use residues to evaluate $\int_0^\infty \frac{1}{1+x^5} \, dx$. Since the intergral goes from $0 \to \infty$ and not $- \infty$ to $\infty$, am I allowed to integrate over a quarter-circle ...
user avatar
1 vote
1 answer
36 views

Contour integral of Singularities

Consider a finite set of distinct real numbers $E_1,E_2,...,E_n$. Let $I$ denote a subset of $1,...,n$ and let $\Gamma$ denote a contour in $\mathbb{C}$ which contains out $E_i,i\in I$ and none of the ...
user avatar
  • 2,398
2 votes
1 answer
31 views

Proving an analytic function in a domain that is not simply connected has an antiderivative

Let $f$ be analytic in $\mathbb{D}$ s.t $f'(0)=0$. Prove that $\frac{f(z)}{z^2}$ has an antiderivative on $\mathbb{D}\setminus\{0\}$. My attempt: Let $\gamma$ be a closed curve in $\mathbb{D}$. If $0\...
user avatar
2 votes
1 answer
72 views

Help in understanding a clever proof of Rouché's theorem

Theorem: $f, g$ are analytic in a region $\Omega . C$ is a circle/rectangle/(simple closed curve) which along with its interior is contained in $\Omega$. Suppose that $|f|>|g|$ on $C$. Then $f$ and ...
user avatar
4 votes
1 answer
177 views

Evaluation of integral with double arctan

How can we evaluate this integral? $$\int _0^{\pi }\arctan \left(\frac{\sin x}{a+\cos x}\right)\arctan \left(\frac{\sin x}{b+\cos x}\right)dx$$ I have solved this problem for $a,b\ne (-1,1)$ my result ...
user avatar
1 vote
0 answers
59 views

How can I evaluate this contour complex integral?

Question: $$\oint_{c}^{} \frac{e^{{z}^{2}}}{z-2}dz$$ And the contour is this figure: Now it would have been great if they had defined the contour but no such luck. The only hint is that I can assume ...
user avatar
5 votes
3 answers
109 views

How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
user avatar
  • 8,526
1 vote
0 answers
30 views

How to set up Contour Integration

So I am evaluating the integral $\int_0^\infty\frac{z^2dz}{(z^2-4)(z^2+9)}$. The residues for the four poles at $\pm2,\,\pm3i$ give$$\operatorname{Res}_{\pm2}(f)=\pm\frac{1}{13},\,\operatorname{Res}_{\...
user avatar
0 votes
0 answers
32 views

Interchange of differentiation and integration sign.

Let $f:\Omega\to \mathbb C$ be a function (where $\Omega$ is an open set) and suppose $\phi:\mathbb C\times \Omega\to \mathbb C$ be a function such that $f(z)=\int_C \phi(\zeta,z)d\zeta$ where $C$ is ...
user avatar
0 votes
1 answer
62 views

Complex integral $\int_{1-i}^{1+i}|z|^2 dz$

Question: Find $\int_{C}|z|^2 dz$, where $C$ is the line segment from $1-i$ to $1+i$. My attempt: We substitute $z(t) = 1 + it$ for $-1\leq t\leq1$ with $dz/dt = i$. The integrand becomes $|z(t)|^2 = ...
user avatar
6 votes
3 answers
383 views

Application of residue at infinity

I am trying to figure out how to find a contour to solve for this integral $$ \int_{-\infty}^{\infty}\frac{2x}{x^2+x+1}dx = -\frac{2\pi}{\sqrt{3}} $$ using the residue theorem and the residue at ...
user avatar
10 votes
1 answer
336 views

Evaluate $\int_{0}^{\infty} \ln(1+\frac{2\cos x}{x^2} +\frac{1}{x^4}) \, dx$

Numerical evidence suggests the following $$I=\int_{0}^{\infty} \ln\left(1+\frac{2\cos x}{x^2} +\frac{1}{x^4}\right) \, dx\stackrel{?}{=}4\pi W\left(\frac{1}{2}\right)=4.42\cdots$$ where $W(x)$ is ...
user avatar
  • 3,092
0 votes
0 answers
30 views

How to evaluate the following contour integral?

I have recently studied Cauchy's integral formula which states that if $f:\Omega\to \mathbb C$ be a holomorphic function and $C$ be a positively oriented simply closed curve whose interior is also ...
user avatar
1 vote
0 answers
28 views

Is this a valid application of Jordan's lemma?

Suppose I'm trying to evaluate the following integral along a large semicircle in the upper half plane: $$ \lim_{R\to \infty} \int_{C_R} e^{iaz} \, f(z) \, , \qquad \text{where } f(z) = \frac{1}{z} \, ...
user avatar
1 vote
1 answer
47 views

Computing a contour integral for ranges of $r$

Compute $$\int_{|z|=r}\frac{e^{\sin(z^2)}}{(z^2+1)(z-2i)^3}\;dz$$ when $0<r<1,1<r<2$ and $r>2$. My attempt: For $r<1$, the integrand is holomorphic and by Cauchy-Goursat the ...
user avatar
  • 2,340
0 votes
0 answers
25 views

Question about the phase difference along a keyhole contour

I am trying to perform an integration along a keyhole contour where the integrand involves $\log z$ and $\log(1+\sqrt{1+z^2})$. Choosing the principal branch for the functions $\log z$ and $\sqrt{1+z^...
user avatar
2 votes
0 answers
51 views

Inverse Mellin transform of products of gamma functions

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
user avatar
  • 31
0 votes
1 answer
28 views

integration over a circle using residues

We are asked to solve the following integral using residue theorem $\int_{|z|=1} \frac{1}{z\sin^2z} dz$. I was able to show that it has one pole of order 3 inside $|z|=1$ given by $z=0$. We know that $...
user avatar
2 votes
1 answer
93 views

Finding the value of $\frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) d\theta$

I wanted to find the value of $$\frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) d\theta:(n \in \mathbb{N} )$$ I believe it is the real part of the following integral: $$\frac{1}{2\pi} \...
user avatar
  • 414
1 vote
1 answer
22 views

Do holomorphic bijections map contours enclosing the origin (winding number $1$) to contours enclosing the origin (with winding number $1$)?

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\g}{\gamma}$So I was recently reviewing the proof of Lagrange-Burmann inversion: Consider functions of the form $f:\Bbb C\to\Bbb C,\,w\mapsto w/\psi(w)$ ...
user avatar
  • 8,526
0 votes
1 answer
29 views

Residues with Inequalities

If $f$ is an entire function such that $|f(z)|\leq A|z|$ for all $z\in\mathbb{C}$ for some fixed $A>0$, then can I write $$\left|\frac{1}{2\pi i}\int_\gamma\frac{f(z)dz}{(z-c)^n}\right|\leq\frac{1}{...
user avatar
  • 29
1 vote
0 answers
16 views

Integral Representation for the Legendre Function of the First Kind

In the text Special Functions and their Applications by N.N. Lebedev (pp.172-173), a derivation is presented for a certain integral representation of the Legendre function of the first kind $P_{\nu }(\...
user avatar
4 votes
0 answers
48 views

Integral of the Laurent series with Bessel function coefficients

I'm trying to integrate a function of the following form, where $A$ and $B$ are both positive: $$\int_0^{\infty}\exp\left(\frac{A}{2}\left(\frac{B}{1+x^2}-\frac{1+x^2}{B}\right)\right)dx.$$ My first ...
user avatar
0 votes
1 answer
32 views

Contour integral with branch cut and cubic root

I am having fun with contour integration and I try to compute the integral $$\int_0^\infty \frac{z^{1/3}}{z^2+\pi^2}dz$$ This integral is equal to $\pi^{1/3}/\sqrt{3}$. However I cannot obtain that ...
user avatar
1 vote
0 answers
42 views

Solving contour integrals

When evaluating the integral of real functions like $\int_0^\infty f(x)\cos(ax)dx$ where $a$ is a real constant, why can't I simply do contour integration by using the complex function $f(z)\cos(az)$? ...
user avatar
1 vote
1 answer
54 views

Equality of contour integrals in Stein & Shakarchi's of Prime Number Theorem

I am studying the proof of the prime number theorem in Stein & Shakarchi's Complex Analysis, and am not sure about the justification of a particular equality. The equality I'm interested in is ...
user avatar
2 votes
0 answers
45 views

Maps that preserve winding numbers

Update: now crossposted to MathOverflow: https://mathoverflow.net/questions/419705/maps-that-preserve-winding-numbers I am looking for a characterisation of the continuous maps on some subset of $A\...
user avatar

1
2 3 4 5
71