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

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2
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1answer
55 views

Finding an upper bound for $|\int_{\gamma}e^{1/z}dz|$.

Find the upper bound for $|\int_{\gamma}e^{1/z}dz|$ where $\gamma$ is the part of the circle $|z| = \sqrt{8}$ from $2+2i$ to $-\sqrt{8}$. This is a question my professor went through quickly during ...
1
vote
3answers
141 views

How to compute the following integral involving hyperbolic functions?

I'm thinking about contour integrals, but I'm not sure. Thanks for your attention :) $$ \int_{0}^{\infty} \frac {\sinh ax \sinh bx}{\cosh cx} dx \\a,b,c \in \Re $$
1
vote
1answer
72 views

Manipulation of Cauchy's Integral Formula

$\quad$ Using Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|\lt1$ in terms of a contour integral around the unit circle, $\zeta=e^{i\theta}$. $\quad$ ...
0
votes
2answers
57 views

Complex Analysis, Integral over a Square

Given that $C$ is the boundary of the square with corners at $\pm4 \pm4i$ (sorry my formatting always seems to be stubborn, but that is plus or minus 4 plus or minus 4i, I am asked to compute $$\int_C ...
1
vote
1answer
35 views

A problem with Cauchy Theorem

I want to resolve the folowing contour integral, using the Cauchy theorem: $$ \oint_C \cot(\pi z)\,dz $$ where $C$ is rectangle defined by $x=\frac{1}{2},x=\pi, y=-1, y=1 $ I do understand that ...
0
votes
2answers
59 views

Contour Integral with Real Singularity

So, if I'm working in spherical coordinates, how would I evaluate the following integral? I know that I'm supposed to use contour integration and Jordan's lemma, but the fact that the singularity is ...
1
vote
1answer
43 views

Wallis' Formula

How can I show the following, for $n\geq 0$: $$ \frac{1}{2\pi} \oint_{\ \Gamma} \frac{1}{z} \left(z + \frac{1}{z}\right)^{2n} dz $$ using a contour $\Gamma$ defined as the unit circle centered at ...
1
vote
1answer
41 views

Residue of Function - Cauchy's Theorem

I need to find the poles and residues of the following function: $$f(z) = \frac{1}{az+\frac{1}{2}b(z^2+1)+\frac{c}{2i}(z^2-1))}$$ which can be rewritten (or at least I have) as: $$f(z) = ...
1
vote
0answers
23 views

Different methods of calculating $\zeta(s)$'s Laurent series.

Initially, I thought that calculating$$\int_\gamma \frac{\zeta(z)}{(z-1)^n}dz$$ directly, where $n \in \mathbb{Z}$ and $\gamma$ is an anticlockwise contour around $z=1$ with winding number $1$, would ...
0
votes
1answer
36 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
1
vote
0answers
98 views

integrate a difficult function

I can't solve it. please help! I tried everything. Integration by parts - doesn't work. but maybe I didnt do it right. I tried to substitute , but I'm stuck. $$\int \frac{x}{\cos x}\sin(\tan ...
3
votes
1answer
116 views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Prove the following using complex analysis $$\tag{1}\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx=\frac{\pi}{2}\ln(2)$$ I found this problem in Schaum's outlines of complex variables. I thought that we ...
14
votes
3answers
330 views

Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
7
votes
4answers
247 views

Integral $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
2
votes
0answers
85 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$ [duplicate]

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
14
votes
1answer
403 views

Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
1
vote
3answers
60 views

How to handle the complex integration of this function around a branch point

I have this complex integral to which I don't know if it's possible to assign a value: The integral is on a small circle around the origin. The function is $\frac{1}{(z-1)\sqrt{z}}$. The fact is ...
1
vote
1answer
74 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
0
votes
1answer
103 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
3
votes
1answer
99 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
2
votes
1answer
98 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)\mathrm d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^n\mathrm d\theta $$ where $n$ is a non-negative integer. This problem is ...
0
votes
0answers
56 views

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ ...
2
votes
1answer
75 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
3
votes
1answer
183 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
2
votes
2answers
169 views

Integrate $I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx$

Hi I am trying to evaluate the integral $$I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx.$$ Note we can write the integrand as $$ x^2 \ln\big(\frac{e^x-e^{-x}}{2}\big) ...
1
vote
1answer
48 views

complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
1
vote
1answer
59 views

Contour integral of $\frac{\bar{z}}{z-Z}$ on a square centered at the origin

I am having trouble calculating the following integral: $\oint_C \frac{\bar{z}}{z-Z} dz$ Here, Z is a complex constant and C is the contour of a square of side $2a$ centered at the origin. I ...
2
votes
1answer
167 views

Integrate $\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta$

Hello I am trying to integrate $$ I=\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta $$ which is similar to Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
2
votes
2answers
197 views

Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
0
votes
1answer
68 views

Evaluating Contour Integral

How do I go about evaluating the following by contour integration? $$ \int^1_0 \frac{dx}{(x^{2} - x^{3})^{1/3}} $$ The question does not fit in the standard form of : $\int^{2\pi}_0$ or ...
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
93 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
45 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
119 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
336 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
23 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
85 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
22 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
24 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
89 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
148 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
188 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
31 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
171 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
108 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
80 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
111 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
57 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 ...