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

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0answers
35 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
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0answers
58 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
103 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
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1answer
39 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 ...
6
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2answers
236 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
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2answers
106 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
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1answer
101 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
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2answers
148 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 ...
0
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0answers
33 views

Contour integration!Help

I have to integrate a function following the route from the point $(0,0,0)$ to $(1,1,1)$ which consists of the 2 curves $C=(t,t^2,0)$ and $K=(1,1,t)$ $0\leq t\leq 1$ .Is it right to take the 2 ...
1
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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
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1answer
51 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 ...
4
votes
2answers
162 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
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2answers
77 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
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0answers
166 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
117 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
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2answers
382 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
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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
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2answers
174 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
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0answers
44 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
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1answer
134 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
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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 - ...
3
votes
2answers
114 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
2
votes
1answer
88 views

Why does the Cauchy-Goursat theorem not apply here?

Let $C$ denote the positively oriented boundary of the half disk $0 \le r \le 1, 0 \le \theta \le \pi$, and let $f(z)$ be a continuous function defined on that half disk by writing $f(0) = 0$ and ...
6
votes
2answers
185 views

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Integrate $$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi. $$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
5
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3answers
220 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
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0answers
42 views

Evaluating this complex integral, how?? [duplicate]

Looking through past papers to prepare for my exam and found this: You are asked to integrate this around the following contour: and show that it's equal to: I have found the residue of this ...
0
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1answer
39 views

Cauchy Riemann equations, do these satisfy it??

I have this question and am unsure of my approach. I have applied the Cauchy Riemann conditions to it: and found that this condition is true. Is that sufficient and does it make sense?
1
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1answer
75 views

definiteinteggral

The integral is given by $$\int_0^1 \frac{\ln (1-x)\ln x}{1+x} dx = \frac{1}{8}\big(-\pi^2\ln(4) +13\zeta(3)\big).$$ Any ideas how to prove? We cannot solve the integral so easily because we cannot ...
0
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1answer
111 views

Find the residue(s) of this function at each pole that lies in the contour?

Going through past papers and found this residue question I can't do. The question asks you to find the residue at each pole that lies in the contour shown. I've got as my answer for the poles ...
0
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1answer
59 views

Complex integrals over an ellipse instead of circle?

I was looking through past papers and found this integral: Which should be evaluated over an ellipse with I've done these plenty of times over a circle with |z| = 2 etc, but where do I start in ...
1
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0answers
118 views

evaluate the integral $I =\int_0^{+\infty} e^{ix^2}dx$

"Evaluate the integral $I= \int_{0}^{\infty} e^{ix^{2}}\, dx$. Let R > 0 and consider the closed contour $C_R = C(1)_R + C(2)_R + C(3)_R$ where $C(1)_R$ is the segment of the positive real axis from ...
4
votes
1answer
113 views

1 dimensional integral, definite integral.

Trying to show $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1-x}dx=-\frac{1}{4}\pi^2 \ln(2)+\zeta(3). $$ I am unsure how to approach this integral as I do not know how to use a power series representation ...
4
votes
5answers
257 views

Integral, definite integral

How can we prove $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}? $$ This has been one of the integrals that came out of an integral from another post on here, but no solution to ...
2
votes
1answer
92 views

Integral, 1 dimension

Re-doing last post since it was incorrect and corrected by many people on here. $$ \int_0^1 \frac{\tanh^{-1}(x)\ln x}{x(1-x^2)}dx $$ I have tried substitutions since $\tanh^{-1}(x)$ and $1-x^2$ are ...
0
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0answers
83 views

Integral Involving Trigonometric Functions and Exponential (Related to Marcum Q-function)

I want to solve this integral $$ \int_{0}^{\infty}\int_{0}^{2\pi}\exp(-ar^2)\exp(r\,b(\cos\theta+\sin\theta))r^{m}\cos^{m}(2\theta)d\theta \,dr,$$ where $a$ and $b$ are constants. I know how to ...
0
votes
1answer
99 views

Using residue theorem to solve double integral, involving singular roots?

I am a physics grad student (high energy), I've come across a problem while doing a certain loop integral that I don't understand. I've removed as much of the physics as I can so that this is just a ...
4
votes
1answer
173 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
1
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1answer
66 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
0
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1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
3
votes
1answer
62 views

Find the analytic continuation of the $ f(z) = \int_{0}^{\infty} \frac{exp(-zt)}{1+t^2} dt$

Find the analytic continuation of the function $f(z)$ defined by $ f(z) = \int_{0}^{\infty} \frac{\exp(-zt)}{1+t^2} dt$ , $ |\arg(z)| < \pi/2$ to the domain $ -\pi/2 < \arg(z) < \pi$ I ...
1
vote
1answer
390 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
5
votes
3answers
230 views

How to compute $I_n=\int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}$?

I'd like to compute: $$ I_n = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}. $$ We have, quite easily: $$ I_0 = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{1}{\cosh^2 x}=\left[\tanh ...
4
votes
1answer
115 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
3
votes
0answers
66 views

Branch-point order and Cauchy representation

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. Suppose we have the following representation: ...
11
votes
1answer
321 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$ \int_{0}^{\infty} {\cos\left(\pi x^{2}\right)\over 1 + 2\cosh\left(\,2\,\pi\,x\,/\,\sqrt{\,3\,}\,\right)}\,{\rm d}x ...
0
votes
1answer
50 views

Finding the complex integral along an arc

How can we evaluate complex expressions like these$\int_C(Z-Z^2)dZ$ where $C$ is the upper half of the circle $|Z-2|=3$
0
votes
1answer
124 views

Very difficult contour integral

I have to compute this integral and I don't have any idea how to get further on: $$\frac{1}{2 \pi i} \int_{\mid z \mid = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz$$ I tried Rouché to maybe ...
6
votes
1answer
145 views

Line contour integral of complex Gaussian

Say I have the entire function $$f(z)=e^{-\frac{1}{2}z^2}.$$ I would like to consider the integral $$I=\int_\Gamma f(x)dz,$$ where $\Gamma$ is a line with negative slope $<1$ in $\mathbb{C}$ (so if ...
6
votes
1answer
115 views

Contour integration of a meromorphic function

Given a meromorphic function $f$ which is uniformly bounded on the upper half plane. Assume that $\int_{-\infty}^{+\infty} f(x)dx$ is absolutely integrable. Then Cauchy's integral theorem suggests ...
2
votes
1answer
317 views

Integrating $\int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$

Let me quote the passage from the book, and then I'll explain the notation. Let us integrate $$ (i) \ \int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$$ $$ (ii) \ \int ...