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

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4
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55 views

Integral $\int z^2\Re(J_1(z))dz$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}\right]dz $$ where $a\in \mathbb{C}$ and ...
6
votes
1answer
100 views

Is there a deep reason why replacing $\cos(x)$ with $e^{ix}$ and taking the real part often makes a contour integral work out?

I'm grading a complex analysis course right now and it turns out to involve a lot of contour integration. For instance, students are asked to find the integral $$\int_0^\infty \frac{\cos (ax)}{(x^2 ...
2
votes
0answers
36 views

Integral using Cauchy's integral formula and residue theorem

So, I'm having trouble getting the correct value for the integral $\int_0^{2\pi} \frac{\cos^2(3\theta)}{5-4\cos(2\theta)}\mathrm{d}\theta$. I substitute the exponential form of cosine into the ...
10
votes
1answer
125 views

why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...
2
votes
1answer
46 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
2
votes
1answer
61 views

What is the solution to this integral?

In some calculation, I encounter an integral of the form \begin{equation} \int_{-\infty}^\infty \text dz\ \frac{1}{z-i\varepsilon}e^{- a z^2+i b z}, \end{equation} where $a>0$ and $b$ are some ...
10
votes
1answer
192 views

Physical interpretation of residues

What is physical interpretation of residues of poles (of any order) of a complex function? Poles represents the points where a complex function cease to be analytic and residues are calculated to ...
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vote
0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
2
votes
3answers
64 views

A question about complex integration of $\frac{1}{p(z)}$

Let $p(z)$ be a polynomial of degree $n\ge 2$. Is it true that, there is a $R>0$ such that $$\int\limits_{|z|=R}{\frac{1}{p(z)}dz}=0?$$ My attempt is: there is a $R>0$ such that $|p(z)|\ge ...
1
vote
1answer
30 views

Integration of Bessel functions:Finding a suitable contour

I have below function to integrate; $$\int_{0}^{\infty} \frac{J_{0}(ax)x^3}{k^2-x^2} dx$$ here $a,k$ are constants. Since this is an odd function, I am not allowed to extend the limits from negative ...
0
votes
0answers
50 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
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vote
0answers
19 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
2
votes
1answer
42 views

Residue theorem on even function integration

I need to integrate below function; $$\int_{-\infty}^{\infty} \frac{\sin(pR)}{R}\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. Since this is an even function of $p$, I tried applying the residue ...
5
votes
4answers
133 views

calculate $\int_{0}^{\pi} \frac{dx}{a+\sin^2(x)} $using complex analysis

where $a>1$ calculate $$\int_{0}^{\pi} \dfrac{dx}{a+\sin^2(x)}$$ I tried to use the regular $z=e^{ix}$ in $|z|=1$ contour. ($2\sin(x) = z-\frac1z)$, but it turned out not to work well because ...
3
votes
0answers
85 views

Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where ...
2
votes
0answers
40 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
1
vote
0answers
24 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
6
votes
2answers
111 views

Is this a Morera´s Theorem Application?

Let $G \subset \mathbb C$ be a domain and $f: G \to \mathbb C$ a continuous function such that for any closed and rectifiable path $\gamma \subset G$, $$ \left| \oint_\gamma f(z)dz \right|\leq \left( ...
3
votes
1answer
52 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
0
votes
1answer
34 views

Calculate complex integral

Let $C$ be a circle $\gamma=\partial B (0,2)$ oriented positively. I have to calculate $$\int_\gamma \frac{-\cos(1/z)}{\sin(1/z)z^2}dz$$ My attempt: Notice that $\sin(1/z)$ is meromorphic inside ...
2
votes
2answers
47 views

Laplace Inverse

I want to find the laplace inverse of $$s^{-3/2}$$ the steps given in the solution manual are as follows: $$\frac{2}{\sqrt\pi}\frac{\sqrt\pi}{2s^{3/2}}=2\sqrt{\frac t\pi}$$ I know the first part ...
0
votes
0answers
38 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
1
vote
1answer
64 views

$\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

It's fairly straight forward to show that $$ \int_0^{\infty} \exp(it^2) dt = \frac{\sqrt{\pi}}{2}\exp\left(i\frac{\pi}{4}\right) $$ via complex contour integration over a contour shaped like a piece ...
6
votes
6answers
278 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
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0answers
30 views

Series involving Laguerre polynomials

Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = ...
2
votes
1answer
53 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
1
vote
1answer
69 views

What enclosure should I choose to evaluate the next integral using residues?

I have to solve the next integral: $$\int_{-\infty}^{\infty} e^{ibx}(e^{ia/x}-1)dx$$ where $a,b$ are real parameters. I can use Jordan´s Theorem to show that as $f(z)=e^{ibz}g(z)$ where $g(z)=(e^{ ia ...
3
votes
0answers
51 views

Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
0
votes
2answers
51 views

Contour Integration of a Complex function

In the context of contour integration: For positive real values of $\alpha$ the following integral is $$ I(\alpha)=\int_{-\infty}^{\infty}\frac{e^{it\alpha}}{1+t^2}dt=\frac{\pi}{e^{\alpha}} $$ Why ...
2
votes
2answers
65 views

evaluate $\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx $

This is supposed to be a very easy integral, however I cannot get around. Evaluate: $$\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx$$ What I did is: $$\int_{0}^{2\pi}\frac{dx}{\cos x + \sin x ...
5
votes
0answers
70 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
0
votes
0answers
50 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
0
votes
1answer
30 views

Contour Integration of Simple Closed Contour

Let C be any simple closed contour inside the annulus 4 < |z| < 6. Show that there holds: $$ \int_C \frac{dz}{z^2+1} = 0$$ To begin: I know that there are poles at $\pm i$ and that the ...
2
votes
3answers
60 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
3
votes
1answer
27 views

Complex contour integral of fraction of polynomials

Let $n \in \mathbb{N}_0$ and set $p(z) = z^n + a_1 z^{n-1} + \cdots$ and $q(z) = z^{n+1} + b_1 z^{n} + \cdots$ to be two monic complex polynomials with no common zeros. I want to prove that ...
1
vote
2answers
22 views

Locations of singularities of a function with respect to given contours

Show that $\int_{C_1}f=\int_{C_2}f$, where $C_1:|z|=1$, $C_2:|z|=2$, and $f(z)=\frac{2z+1}{\sin z}$. Hint: Locate the singularities of $f$ in each case and indicate their location with respect ...
0
votes
1answer
34 views

Complex Integrations and Contours

Show that $$\int_C \frac{2 z^2-5}{(z^2+1)(z^2+4)} dz \le \frac{\pi R (2 R^2+5)}{(R^2-1)(R^2-4)} $$ Let $C$ be the upper half of the circle $z=R$ for any $R>2$. Do I need to actually find the ...
0
votes
0answers
53 views

Prove: $\int_C \frac{dz}{z^2+1} =0$ on the annulus $6\lt |z| \lt 8$

Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds: $$\int_C \frac{dz}{z^2+1} =0$$ This has two singular points, $z=\pm i$, these are ...
4
votes
1answer
39 views

$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$ on an ellipse

I am a self-studier and this is a problem from a course I've been doing. I would appreciate help showing: $$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$$ ...
0
votes
2answers
51 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
2
votes
2answers
41 views

Calculating residue $\int_C \frac{8-z}{z(4-z)}dz$

I want to calculate the following: $$\int_C \frac{8-z}{z(4-z)}dz$$ $C$ is a circle of radius $7$, centered at the origin,negative oriented. I want to do this via finding the residues at $z=0,4$. I ...
3
votes
4answers
96 views

Integral using contour integration [duplicate]

Here is the integral I want to evaluate: $$\int_{0}^{2\pi} \frac{dx}{a+b \cos x }, \quad a>b >0$$ Apparently there are limitations as to what values $a, b$ are supposed to take but let us not ...
3
votes
1answer
76 views

Steepest descent method with movable maximum

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral $$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$ where $C$ is some contour in the ...
3
votes
1answer
35 views

Polynomial Inequality via Contour Integration

Problem. Let $P(z)=\sum_{k=1}^{n}a_{k}z^{k}$ be a polynomial which is real on the real axis. Prove the inequality ...
3
votes
3answers
72 views

Evaluating past exam problem: $\int_C \frac{\sin z}{(z+1)^7} \mathrm{d}z$

I want to evaluate the following: $$\int_C \frac{\sin z}{(z+1)^7} \mathrm{d}z$$ Where $C$ is the circle of radius $5$, centre $0$, positively oriented. Now this has one root at $z=-1$. Now I should ...
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vote
0answers
21 views

real integral using residue theorem

Edit before posting: my result didn't match with the solution, found the error while posting, figured I would post it anyway because someone else might find it useful I'm trying to solve: ...
3
votes
1answer
74 views

How to evaluate such integral with pole structure?

Let's have integral: $$ I = \int \limits_{-\infty}^{\infty} \frac{e^{-\frac{x^{2}}{2}}}{x - a - i0} $$ How to evaluate it? I tried to do following: $$ \frac{1}{x -a - i0} = \int ...
3
votes
2answers
80 views

How to integrate $e^{-\cos(\theta)}\cos(\theta + \sin(\theta))$

I am struggling to find a way to evaluate the following real integral: $$\int_{0}^{2\pi}e^{-\cos(\theta)}\cos(\theta + \sin(\theta))\:\mathrm{d}\theta$$ The exercise started by asking me to ...
4
votes
3answers
112 views

Evaluating $\int_{-\infty}^{\infty}\frac{\cos x}{e^x + e^{-x}}$ using the Residue Theorem

I consider the complexification $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Poles of $f$: $\text{Denominator}=e^{-z}(e^{2z}+1)=0\Rightarrow e^{2z}=-1=e^{i(\pi + 2\pi k)}\Rightarrow z=\frac{i\pi(1+ 2k)}{2}$, ...
1
vote
0answers
23 views

Question about an boundary integral equation with a jump in the boundary

I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions. Applying Green's second identity the representation formula can be derived: ...