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

learn more… | top users | synonyms

0
votes
1answer
56 views

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$. Isn't $z^n-z^{mn}$ analytic, so the value is 0? And if it isn't analytic, letting $z(t)=2e^{it},t\in[0,2\pi]$ gives ...
1
vote
1answer
34 views

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$ [duplicate]

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$. Is there a special trick to solving this? I tried letting $z(t)=2e^{it},t\in[0,2\pi]$, which gave me ...
1
vote
0answers
27 views

Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Let $m\in\mathbb{N},t>0$ how to compute the integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$, where $\gamma$ is contour $\{|\arg(\mu+1)|=\pi/4\}$ transversed upward? Here are my ...
3
votes
1answer
70 views

Cauchy Principal Value of $\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$

The problem here is to evaluate $$ \int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx $$ for $a,\mu >0.$ Clearly this integral doesn't converge in the usual sense, but we can calculate its ...
1
vote
1answer
36 views

Higher order derivatives of Cauchy integral

An excerpt from my course notes in introductory complex analysis. Given a curve $\Gamma$ in the complex plane and a continuous function $\varphi : \Gamma \to \mathbb{C}$, define its Cauchy integral ...
8
votes
2answers
216 views

How to evaluate this integral $\int_{0}^{\infty }\frac{\ln\left ( 1+x^{3} \right )}{1+x^{2}}\mathrm{d}x$

How to evaluate this integral $$\mathcal{I}=\int_{0}^{\infty }\frac{\ln\left ( 1+x^{3} \right )}{1+x^{2}}\mathrm{d}x$$ Mathematica gave me the answer below $$\mathcal{I}=\frac{\pi }{4}\ln ...
0
votes
1answer
76 views

Possible to integrate $\ln |x-y| dy$ on a circle? [closed]

I am wondering what is the solution to $$\int_{\partial D}\ln |x-y| dy$$ when $D$ is the unit disc in $\mathbb{R^2}$ and $x \in \partial D$. Is this even possible analytically?
0
votes
0answers
29 views

Contour integral of 1/z

Consider the integral $\int_{-\infty}^\infty dx \frac{1}{x+i \epsilon}$, where $\epsilon$ is a positive real number. We can evaluate the integral by closing the contour in the complex plane and then ...
1
vote
0answers
29 views

Solving differential equation by transform with contour integral

In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
0
votes
0answers
37 views

Integrals that don't coincide with the Riemann integral?

This is probably a lame question, but I was wondering if there exist integrals that do not coincide with Riemann's integral for function that are integrable with respect to both these integrals?
4
votes
1answer
55 views

Cauchy Residue Theorem Integral

I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta $$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform ...
0
votes
0answers
38 views

Contour Integration of Definite Integral of Sine and Cosine, 4th order pole

There is this problem from Complex Variables (Brown and Churchill) regarding the definite integral of a sine and cosine function. It's supposed to be integrated using residues. I've been trying to ...
0
votes
1answer
43 views

Complex integral of the function $f(z)=\dfrac{1}{z^4+1}$

I must calculate this integral $$\int_C\dfrac{dz}{z^4+1}$$ , where $C$ is the circle $x^2+y^2=2x$. My result is $\int_C\dfrac{dz}{z^4+1}=-\dfrac{\pi}{\sqrt{2}}$ , but my book "A collection of problems ...
3
votes
1answer
278 views

Compute the series $\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$

I need to compute $$\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$$ This an exercice of "Amar and Matheron, complex analysis". I proved the convergence and now to compute the sum, I follow the ...
2
votes
3answers
183 views

Evaluation of Bose-Einstein and Fermi-Dirac Integrals

Can anyone help me with the following integral, which has arisen from my study of cosmology? $$\int_0^\infty\frac{p^2dp}{e^{(p\pm\mu)/T}+ 1}$$ I have tried substitution and using standard integral ...
3
votes
1answer
57 views

Proving a modified version of Jordan's lemma?

I want to prove the version of Jordan's lemma which say's that: Around a contour as shown below: The integral: $$\int_{\Gamma} e^{az}f(z)dz$$ will go to $0$ if $$|f(z)|\le \frac{M}{R^\alpha}$$ for ...
3
votes
2answers
90 views

How to find $\int_{-\infty}^\infty \exp(-x^2) \, dx$ with contour integration?

Contour integration is a very powerful tool. But what if a function has no poles or zero's ? For instance : How to find $\int_{-\infty}^\infty \exp(-x^2) \, dx$ with contour integration?
0
votes
1answer
54 views

The complex integral of the reciprocal of polynomial is constant on sufficiently large circles

Let $P$ be a polynomial of degree $n ≥ 2$. Suppose all the roots of $P$ lie in the disk $D_{r}(0)$. Let $R > r$ and $$I(R)=\int_{|Z|=R}{1\over P(z)}dz$$ Prove that the integral is constant. I ...
0
votes
1answer
31 views

Am i evaluating this contour integral correctly?

Let $\gamma$ be the circle of radius 2 centered at the origin. $$ \int_{\gamma} \frac{1}{z^2+i}dz $$ I tried factoring the denominator out to where $ \int_{\gamma} \frac{1}{z^2+i}dz $ = $ ...
0
votes
1answer
24 views

Algebraic confusion

How does $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack k + \frac{1}{2i}\left( z - \frac{1}{z}\right) \rbrack^2}$$ become $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack \frac{1}{2iz} \left( ...
0
votes
0answers
27 views

Two different answers with applying the Cauchy integral formula and parametrizing.

I have the integral $\int_{|z|=1}\frac{cosz}{z}dz$ By applying the Cauchy Integral Formula, I get that this equals $2\pi i*cos(0) = 2\pi i$ Is this correct? If I parametrize the integral with $z= ...
5
votes
1answer
122 views

Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
1
vote
0answers
41 views

Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
1
vote
1answer
46 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
6
votes
2answers
146 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...
2
votes
3answers
44 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
1
vote
3answers
46 views

Contour Integration: non-convergent integral

The question is $$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$ My attempt: $$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$ $$I=-\frac{1}{4} \Big[ ...
0
votes
1answer
42 views

Contour Integration with pole on contour

I have come across an example I don't understand.. So, here is the problematic part: Consider the integrals: $ I = \int_C \frac{e^{iz}}{z} dz $ $ J = \int_C \frac{e^{-iz}}{z} dz $ Where $C,C_-, ...
0
votes
1answer
35 views

Generalisation of an already generalised integral

Inspired by these two questions: Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ Interesting integral formula I ask whether the following integral has a closed form: ...
0
votes
0answers
42 views

Integral with logarithmic residue (Residue general method)

I was reading about this post, about Integration in complex plane with logarithmic residue: Integral with logarithm - residue And I thought about: what if $R(x)$ function is inside the logarithm? ...
15
votes
2answers
170 views

How to solve $\displaystyle\int_0^{\frac{\pi}{2}} \ln{(x^2 + \ln^2{(\cos{x})})} \mathrm{d}x$ [duplicate]

$\displaystyle\int_0^{\frac{\pi}{2}} \ln{(x^2 + \ln^2{(\cos{x})})} \mathrm{d}x$ I was given this integral yesterday by someone on a forum and after a few hours of having a go at it I didn't really ...
3
votes
2answers
60 views

Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
0
votes
0answers
32 views

Limit of complex integral with no primitive

I'm having trouble trying to calculate the following limit. I know the answer is not 0, but after several attempts I am stuck on reducing it. We have $z_0$ as a constant complex number and a fixed ...
1
vote
0answers
50 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
6
votes
1answer
174 views

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where ...
2
votes
0answers
115 views

Integration: $\int_0^\infty \frac{\sin(at)}{2\cosh 2\pi t-1} \text{ d}t$

I am trying to calculate the following integral: $$\int_0^\infty \frac{\sin(at)}{2\cosh 2\pi t-1} \text{ d}t$$ The inspiration for it was that a similar integral carries a closed form: ...
2
votes
1answer
44 views

express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
1
vote
1answer
34 views

Normalisation of Lorentzian function proof

According to WolframMathworld, when we normalise the Lorentzian function, we get the following result: $\dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ I'm trying to ...
0
votes
1answer
45 views

Contour integral and primitive

Given $$f(x,y)=\frac{2}{i(1-y)-x}$$ I have to integrate $f$ over the origin-centered circle of radius 4. I see that $$f(x,y)=-\frac{2x+i2(1-y)}{x^2+(1-y)^2}$$ There is a singularity in $-i$ so I ...
1
vote
1answer
33 views

how to choose the path for complex line integrals? (Explicit question)

I have a small question about complex line integrals. I give an example to demonstrate what my question is. Consider the integral $$\int_{|z|=2}\frac{1}{z-1}dz.$$ If I want to calculate this ...
2
votes
0answers
46 views

Integral calculation by using Mellin Transform

I want to use the Mellin Transform (MT) to calculate the integral: $\int_0^{1 } \exp(-2\rho^2) J_0(\pi \rho r)\rho \, d\rho$ in which $r>=0$ and real. I have calculated it by numerical methods. ...
1
vote
2answers
30 views

Given $\vec{F}=(y,x)$, calculate contour integral $\int_C\vec{F}•\vec{dr}$, on length of closed path $C:\vec{r}(t)=(\cos t, \sin t), 0\leq t\leq2\pi$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $c$ and graded ...
0
votes
1answer
25 views

Let $f$ be an arithmetic function of class $C^2$ with $f_{xx}+f_{yy}=0$. Prove that $\oint\vec{F}•\vec{dr}=0$ where $\vec{F}=(f_y,-f_x)$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $b$ and graded ...
2
votes
2answers
79 views

Using contour integrals to evaluate sum - Problem calculating residues

"Compute $$\sum_1^{\infty} \frac{1}{{n^2}{(n+1)^2}}$$ using contour integration" I have used the function $F(z) = \frac {\pi cot\pi z}{z^2(z+1)^2}$ Which has double poles at $z=0$ and $z=-1$ ...
0
votes
0answers
14 views

What exactly is $p$ in numerical contour integration method by Lyness and Delves

http://www.ams.org/journals/mcom/1967-21-100/S0025-5718-1967-0229388-0/S0025-5718-1967-0229388-0.pdf: As far as my reading goes, I do not see how they set $p$ - is it the free variable that can be ...
0
votes
0answers
29 views

Numerical contour integration of a function by Lyness and Delves

In this link, which is one of the first papers dealing with numerical contour integration, in page $12$ right after $(5.12)$, they say: The factor of $4^{-(p^2)}$ in this expression is a number ...
2
votes
2answers
82 views

Contour integration around a closed loop

By considering the contour integral $$\int z^{5/4}(z-1)^{-1/4} dz$$ around a closed loop C that encircles the real interval $[0,1]$, show that $$\int^1_0 x^{5/4}(x-1)^{-1/4} dx = ...
2
votes
1answer
31 views

Residue of pole

I am trying to integrate $$\frac{1}{2\pi} \int^\infty_{-\infty} \frac{6e^{-ipt}}{(p+1)^2 +9} dp$$ I am using a D-contour and I am trying to calculate the residue at the pole $p = -1$. I am trying to ...
3
votes
3answers
146 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
2
votes
1answer
34 views

Contour integral around 'D-contour'

By considering the contour integral aroun a 'D-contour' of the function $$f(z) = \frac{ze^{iaz}}{z^2 + 6 + 25}$$ for $a>0$, show that $$\int\frac{x\sin(ax)}{x^2 +6x +25}dx = ...