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

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0
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2answers
37 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
0
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1answer
20 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
33
votes
4answers
913 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
8
votes
3answers
142 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
1
vote
2answers
22 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
4
votes
0answers
44 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq ...
1
vote
4answers
56 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral ...
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1answer
49 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ ...
1
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0answers
45 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R ...
0
votes
1answer
37 views

Calculating Infinite Real Integrals Using Residues

I want to calculate the following real integral using residues and I am unsure how to proceed. $$\int_{-\infty}^{+\infty}\frac{1- x^2}{1+ x^4} dx$$ I know I must change this to a contour integral so ...
0
votes
0answers
21 views

$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
0
votes
3answers
45 views

$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
1
vote
1answer
28 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if ...
10
votes
0answers
780 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
1
vote
1answer
28 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} ...
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votes
0answers
33 views

An Alternate proof of Nyquist-Shannon

This problem is from Basic Complex Analysis, Part 2A, by Barry Simon. This problem will provide an alternate proof of the strong from of the Nyquist-Shannon sampling theorem (Theorem 6.6.16 of Part ...
1
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1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. ...
3
votes
1answer
65 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
2
votes
3answers
75 views

Improper integral $\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$

How do I solve this? $$\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$$ I know it's a type 3 improper integral, and I'm having issues with these. I think that I need to write it as a ...
1
vote
1answer
24 views

Integral estimation

Is it true that $$\int_{-\pi}^{\pi}\left |\frac{iR^{x}e^{i \theta x}}{1-Re^{i\theta}}\right |\,\mathrm{d}\theta \leq 2\pi R\frac{R^{x}}{1-R}\xrightarrow{R\to\infty}0$$ $$\int_{-\pi}^{\pi}\left ...
0
votes
0answers
11 views

Splitting a contour integral into separate integrals?

Suppose you had a complex contour integral of the form $$ \int_{\alpha + \beta} f(z) \; dz $$ Is this equivalent to $$ \int_{\alpha} f(z) \; dz + \int_{\beta} f(z) \; dz $$ Thanks
0
votes
1answer
42 views

Derive the Fourier Transform

I have been asked to derive the Fourier Transform for $$f(x)=\frac{1}{x^2+a^2}$$ where $a>0$. I know the Fourier Transform is equal to ...
0
votes
0answers
22 views

Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
0
votes
1answer
26 views

Proof of the Primitive Existence Theorem

In the first section of the proof of the primitive existence theorem we are trying to show that $$F\left(z+h\right)-F\left(z\right)=\int_{L_{z,z+h}}f$$ where $\alpha, z,z+h$ are collinear and ...
2
votes
0answers
14 views

Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
0
votes
0answers
17 views

How to prove reflection positivity for $|x|^{-p}$ using Fourier transform (and contour integrals)

This question looks quite lengthy because I'm sketching the proof in the lecture - the two questions (look out for something bold) are actually relatively short. I need some help with a proof in our ...
0
votes
0answers
24 views

Evaluate the following contour integral

Evaluate the integral $$\int_{C}\left(\exp\left(z\right)-\dfrac{2}{z^3}\right)dz$$ where C is any contour from $i$ to $-i$ which does not pass through $0$ I understand that contour integration can ...
0
votes
1answer
15 views

Showing that a function is the restriction of another holomorphic function

Let $f$ be holomorphic on the annulus $\{z:\;1-\epsilon<|z|<1+\epsilon\}$. Define: $$\phi:\;D\to\mathbb{C};\quad \phi(w)=\frac{1}{2\pi i}\int_C \frac{f(z)}{z-w}\,\mathrm{d}z$$ where $C$ is the ...
2
votes
0answers
33 views

$ f(z) = \int _{0} ^\infty e^{-z t^2} dt $ is holomorphic in $Re(z) > 0$

I've been trying to find a reasonable way to approach this but nothing leads to a reasonable result. With $z = a + ib$ the last thing I tried looking at was $$| \frac{f(z) - f(z-h)}{h}| = ...
0
votes
0answers
19 views

conversion from path integral to contour integral

I'm considering the following integral $$I=\int_{-i\infty-\epsilon}^{i\infty-\epsilon}\frac{\psi(v)}{v}\;dv$$ I want to use the following substitution to convert the path integral to a contour ...
10
votes
5answers
395 views

Contour Integration $\int_0^1\frac1{\sqrt[n]{1-x^n}}dx$

I want to compute: $$\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
4
votes
2answers
127 views

Evaluate $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ using a suitable contour

I need to find a good contour for $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ but I don't know which one to choose. Both a semicircular, and rectangular contour look ugly for this. I've been ...
4
votes
1answer
151 views

Show that $\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta = \frac {\pi r^n}{1-r^2}$

I am trying to calculate $$I=\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta$$ where $r\in(0,1)$ I tried substituting $u = e^{2 i \theta}$ and using the Cauchy integral ...
2
votes
1answer
22 views

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it)$

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it),t\in\left[0,\frac{\pi}{4}\right]$. Hint. Use that $\cos 2t\geq ...
4
votes
3answers
591 views

show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

Show that $$\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2\left(\frac{a\pi}{2}\right) $$ I think we can solve it by contour integration, but how? and it would ...
2
votes
1answer
48 views

Evaluate $I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$

This is the last part of a multistage evaluation of $I(x)=\Gamma(x)\Gamma(1-x)$. Through various substitutions we get the integral $$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$$ We're also ...
1
vote
0answers
8 views

Compute an integral involving exterior Riemann mapping

Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. There exists a unique exterior conformal mapping $\Phi$ from ...
3
votes
2answers
1k views

A contour integral with 3 branch points

I came across an exercise in a textbook that says to evaluate $$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \, dx \, , \quad a > b > -1$$ by integrating $f(z) = z^{a-1} ...
0
votes
0answers
19 views

Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
0
votes
2answers
51 views

Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} ...
1
vote
1answer
46 views

Gaussian integral $\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)\,\mathrm d x$ along $[-R,R]+\mathrm i[0,Y]$

Use integration along $\partial Q$ of $Q=[-R,R]+\mathrm i[0,Y]$ to show that for all $Y\geq 0$ it holds that $$\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)~\mathrm dx = \int_{-\infty}^\infty ...
3
votes
1answer
35 views

How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
2
votes
1answer
540 views

Using a keyhole contour

I've noticed that some complex analysis textbooks discuss evaluating real-valued integrals like $\int_{0}^{\infty} \frac{\sqrt{x}}{1+x^{2}} \, dx $ using a keyhole contour before they have defined ...
0
votes
0answers
55 views

Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
0
votes
1answer
23 views

Is integrating $e^{iz^{2}}$, along the real axis in the complex plane the same as integrating the riemann integral of $e^{x^2}$?

In the title, $z\in \mathbb{C}$ and $x\in\mathbb{R}$. More specific to my problem, I am hoping that $\int_{0}^{R}e^{iz^{2}}dz=\int_{0}^{R}e^{x^{2}}dx$. Maybe this is obvious but I want to make sure. ...
1
vote
0answers
21 views

Contour integral along a parabola

The question reads: Evaluate $$ \int_\gamma f(z)dz$$ where $$f(z)=x^2: x,y \in \mathbb{R} $$ and $\gamma$ is the parabola $y=2x^2$ from $x=0$ to $x=2$. This is the first question I've encountered ...
2
votes
1answer
195 views

Choosing between a semicircular contour and a rectangular contour

In a paper I came across (page 10, section 7), the authors state that $\int_{-\infty}^{\infty} \frac{dx}{(b^{2}+x^{2})\cosh ax} $ can be evaluated by "closing the real axis with a semi-circle centered ...
0
votes
1answer
8 views

How would you plot this equation (in the complex plane)?

So I am currently looking at how to calculate the integral of a complex function $f(z)$ within a contour $\gamma$. That is, an integral of the form $$ \int_{\gamma} f(z) \; dz $$ Where the contour is ...
1
vote
2answers
30 views

Mistake while evaluating the gaussian integral with imaginary term in exponent

I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I ...
0
votes
1answer
67 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...