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

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2
votes
1answer
134 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ I_n:=\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old. I am also looking for literature on these integrals ...
1
vote
1answer
61 views

Integration of trigonometric functions times a simple rational function using residues

In the course of my research I have found a few integrals that I would like to have closed-form answers to: $$\int_{c- i \infty}^{c+ i \infty} \frac{1}{z-1} \frac{8 \pi^4 \cot{ \big( \frac{\pi}{6} z ...
1
vote
1answer
86 views

Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$

Hi I am trying to integrate $$ \int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0. $$ These integrals can be ...
1
vote
1answer
44 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
3
votes
0answers
115 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
4
votes
1answer
330 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
0
votes
1answer
21 views

Direction of Contour Integration

When I'm using the residue theorem to evaluate a contour integral, does the simply closed curve always have to be in a counter-clockwise direction? I believe that I can go in a clockwise direction, ...
1
vote
1answer
80 views

$\int e^{\cos(x)} \cos(nx)\ dx$ using the residue theorem

I am trying to evaluate the following integral using the residue theorem: $$\int_0^{2\pi} e^{ \cos(\theta)} \cos(n\theta) d\theta$$ I have already evaluated $\int_0^{2\pi} e^{e^{-i\theta}} e^{i ...
0
votes
2answers
21 views

Parametric equations in complex analysis

I am trying to find $\int_C (1+i-2z')dz$ where$z'$ is the conjugate of $z$ and where C is the parabola $y=x^2$ from $z_1=0$ to $z_2=1+i$. How do I write the parametric equations for this?
0
votes
0answers
22 views

Showing $\int_{\gamma}f(z)dz = \int_{\gamma_1}f(z)dz + \int_{\gamma_2}f(z)dz$ with non analytic points.

Suppose $f$ is analytic on the complex plane except at $z_1,z_2$, that $\gamma_1$ and $\gamma_2$ are simple closed curves with $z_1,z_2$ in their interiors and $\gamma_1$ and $\gamma_2$ are in the ...
1
vote
0answers
83 views

Integrate $ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi$

EDIT/UPDATE: I DO NOT NEED A SOLUTION. SEE SOS440 COMMENT FOR A FULL DETAILED SOLUTION. Hi I am trying to integrate $$ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi, ...
4
votes
2answers
142 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
4
votes
1answer
147 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
1
vote
1answer
30 views

The fourier transformation of complicated function

What is the Fourier transformation of $\operatorname{sech}(at)\operatorname{exp}(bt^2)$, where $a$ and $b$ are some constant?
1
vote
1answer
115 views

Hilbert Transform of cos wt = sinwt.

Hilbert Transform of cos wt = sin wt. Can anyone help me with the proof. in Last Step how this become pi
2
votes
1answer
101 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
3
votes
1answer
79 views

Shifted integral for a Bessel Function

I have an integral of the kind $\int_{-\infty}^\infty e^{- d \cosh(x+i a)} dx $ where $d, a \in \mathbb{R}$. Now, I know that $\int_{-\infty}^\infty e^{- d \cosh{x}} dx = 2 K_0(d)$ and I would ...
0
votes
0answers
31 views

Contour Integral of $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ [duplicate]

I'm trying to evaluate the following integral: $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ $0<a<1$ I've integrated from 0 to $ i\infty $ then from ...
1
vote
2answers
105 views

How does the integral $\int_{D_C} e^{ia z}P(z)/Q/(z)\,\mathrm{d}z$ blow up.

In my book I have a theorem that goes something like the following Let $P(x)$ be $Q(x)$ polynomials such that $\deg(Q) \geq \deg(P) + 2$. Then \begin{align*} \int_{-\infty}^{\infty} ...
0
votes
2answers
54 views

Complex Analysis - Contour Integration

By considering the integral of the function $f(z) = exp(-az^2)$, with $a > 0$, around an appropriate contour, show that the integral $$ I(p,a) = \int_{\infty+ip}^{\infty+ip} exp(-az^2)dz$$ which ...
0
votes
0answers
79 views

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$ With $\sqrt{z}$ I mean the branch with the non-positive real axis as branch cut. With ...
2
votes
2answers
204 views

Problem with Cauchy integrals

Hello everybody I need to solve some integral with the help of the Cauchy Integral Formula (CIF). I'll post near each integral the job that I've done and the question that I can't answer. let $\kappa ...
2
votes
1answer
46 views

Three questions concerning holomorphic functions defined by contour integrals

Consider the following situation: a simple, closed, piecewise smooth curve $\gamma$ in the complex plane and $\Omega$ the bounded connected component of the complement of $\gamma$ in $\mathbb{C}$; a ...
3
votes
2answers
142 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
5
votes
2answers
83 views

Convergence of a line integral along semi-circular arc

There is a line integral in a form, $$\int_\mathrm{arc} \frac{\exp(iz)}{z^2+1} \, dz$$ "arc" is a semi-circular line with radius $R$ on the upper half complex plane. and i know that the integral ...
4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
2
votes
3answers
127 views

Integrate $\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx$

I need to solve for the parameter $\alpha$ after I calculate the integral.$$ \mathcal{R}(\alpha,\beta)=\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx, \ \ \beta >0 $$ The result of this ...
6
votes
1answer
105 views

Integral $I=\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx$

Hello can you please help me solve this integral $$ \int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx=-\frac{2\pi}{3}(\pi^2+24\ln 2). $$ I am trying to work through all logarithmic integrals. Note, ...
10
votes
4answers
308 views

Integrate $ \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$

I am trying to evaluate this integral. $$ I=\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2 $$ Note $$ \ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, \ |x| < 1. $$ I was trying to do ...
11
votes
3answers
372 views

Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$ \int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8} $$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly ...
7
votes
2answers
137 views

Integral $I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $

$$ I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $$ I am trying to solve this integral. This is from the old high school ...
2
votes
0answers
65 views

Typical Contour Inegral Proof

I'm trying to prove that the following contour integral approaches 0 as R -> $\infty$. How exactly would we go about doing this? $$ \int{\log\left(z^{2} + 1\right) \over 1 + z^{2}}\,{\rm d}z\quad ...
0
votes
0answers
50 views

Logarithmic Contour Integration [duplicate]

So, I'm having a really difficult time trying to evaluate the following integral via contour integration (please, no other methods): $$\int_0^\infty{\frac{\log{(x^2+1)}}{1+x^2}} dx$$ Obviously, ...
3
votes
3answers
93 views

Contour integration problem

I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration. Now I used an indented semicircular contour, and the parts lying on the real line and the big arc ...
1
vote
1answer
108 views

Calculation of a Residue

Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find ...
6
votes
2answers
136 views

Integrating $ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $

Compute $$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
1
vote
1answer
61 views

Contour Integration Limitations?

So, I'm trying to evaluate the following integral by complex contour integration ONLY: $$\int_0^\infty{\frac{x^\alpha}{x(x+1)}} dx$$ where alpha is real and not an integer. Obviously, we need to use ...
0
votes
2answers
59 views

Contour Integration Part

I'm trying to evaluate the following integral, and I'm getting stuck on one part. Here's the integral: $$\int_{-\infty}^\infty \frac{\sin(x)}{x(x^2+1)} dx$$ Basically, I'm converting this to the ...
3
votes
1answer
67 views

Integrating around the upper half of $|z|=R$

In a textbook it says that you can show that $ \displaystyle\int_{-\infty}^{\infty} \frac{\cos(x^{2})+\sin(x^{2})-1}{x^{2}} \ dx = 0$ by considering $ \displaystyle f(z) = \frac{e^{iz^{2}}-1}{z^{2}}$ ...
5
votes
4answers
145 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
5
votes
1answer
201 views

Evaluating trigonometric integral using residues

I am trying to evaluate for real positive $\alpha,\beta$ $$\int_{0}^{\infty}\arctan\left(\frac{\alpha}{x}\right)\sin(\beta x)dx$$ using a hint to consider $$\int \log\left(\frac{z+ia}{z}\right) ...
1
vote
2answers
101 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
8
votes
2answers
180 views

Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.

$$ \int_{-\pi/2}^{\pi/2}\cos^m\left(x\right){\rm e}^{{\rm i}n x}\,{\rm d}x ={\pi \over 2^{m}}\, {\Gamma\left(1 + m\right) \over \Gamma\left( 1 + \left[m + n\right]/2\right)\ \Gamma\left( 1 + ...
1
vote
1answer
73 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
3
votes
1answer
165 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
1
vote
0answers
36 views

Show that $\int_{-\infty}^\infty f(t)dt=0$ where $f\in H^\infty(\mathbb{H})$

The problem is stated as follows: Let $\mathbb{H}$ denote the open upper half plane. Let $f \in H^{\infty}(\mathbb{H})$ Suppose $f$ can be extended to be continuous on $\overline{\mathbb{H}}$ with ...
3
votes
0answers
66 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
0
votes
2answers
127 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...
0
votes
1answer
43 views

Is there a lower bound for integration of complex functions?

We are given in our book that the upper bound for complex integrations is $|\int_\gamma f(z)\,dz| \leq mL$ where $L$ is the length of $\gamma$ and $m$ is the $\max(|f(z)|: z\in \gamma)$ and were ...
7
votes
2answers
273 views

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx $$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...