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

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2
votes
1answer
98 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: ...
2
votes
1answer
76 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
30 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
103 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
52 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
203 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
44 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
138 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
72 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
89 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
118 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
98 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
293 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 ...
10
votes
3answers
355 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
130 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
49 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
87 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
103 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
132 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
59 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
143 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
191 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
100 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
179 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
71 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
144 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
62 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
108 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
42 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
261 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 ...
3
votes
2answers
114 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
1
vote
1answer
104 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
3
votes
2answers
153 views

Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$ \int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx $$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
1
vote
1answer
33 views

Integration Exercise.Help!

I have to integrate the function F(x,y)=x+y on the line segment x=t , y=1-t , z=0 from (0,1,0) to (1,0,0) .So what i did is think the line segment as a vector function(curve) σ(t)=(t,1-t,0) with ...
2
votes
1answer
54 views

Determining asymptotic behavior through generating functions

I need to determine the asymptotic behavior of $$a_n=\sum_{k=2}^{n-2}\frac1{\ln k\ln(n-k)}$$ as $n\to\infty$. I know some elementary methods that might help. For example, split the index $\lvert ...
5
votes
2answers
170 views

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral

Calculate $$ I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1. $$ I am trying to solve this integral $I_1$. I know how to solve a related integral $I_2$ $$ I_2:=\int_0^1 \frac{\ln \ln ...
1
vote
2answers
82 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
4
votes
0answers
168 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
6
votes
4answers
119 views

Computing $\int_0^\infty\mathrm{d} x\frac{x}{e^x+1}$ with contour integration

Let's set: $$ \int_0^\infty\mathrm{d}x\frac{x}{e^x+1}=I. $$ I would like to compute it using, presumably, the methods of complex analysis and contour integration.
11
votes
2answers
401 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
2
votes
1answer
33 views

Is there no analytic form of $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ ?

I am trying to find an analytic answer for $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ but it doesn't seem to be in any of the integral tables that I've looked in. I don't think ...
3
votes
2answers
185 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
0
votes
0answers
46 views

Integration using Cauchy Integral Formula

If $f(z)$ is analytic in $|z-a|<R$ and $0<r<R$, then $f'(a) = \frac{1}{r\pi}\int_{0}^{2pi}F(\theta)e^{-i\theta}\,d\theta$, where F($\theta$) is the real part of $f(a+re^{i\theta})$. I ...
0
votes
1answer
144 views

Clarification of Cauchy Principal Value and use of Contour Integration

I am evaluating the improper integral $\int_{-\infty}^\infty{\frac{\sin^3 x}{x^3}dx}$; I am also told to show that this is equal to its principal value, and use this fact to evaluate the integral. I ...
0
votes
1answer
32 views

Closed contour within a closed contour integral

Let $C$ and $D$ be two closed contours, $D$ lying completely within $C$, and let $a$ be a point between $C$ and $D$. Show that: $$ f(a) = \frac{1}{2\pi}\int_{C}\frac{f(z)}{z-a}dz - ...