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

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2
votes
1answer
49 views

Finding a definite integral using complex analysis.

Now, I want to integrate $\int_{0}^{\infty} \dfrac{\cos (2x) -1}{x^2} \mathrm{d}x$, now I attempted to set $f(z)=\dfrac{e^{i2z}-1}{z^2}$ and then integrate around a similar contour to the classical ...
0
votes
1answer
26 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
0
votes
1answer
42 views

Compute $\int_{\gamma} x dz$

Let the perimeter of the square formed by the points $0$, $1$, $1 + i$, $i$ and $z = x + iy$. How can i compute $$\int_{\gamma} x dz$$. Some help to compute this complex integral please.
1
vote
0answers
32 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
2
votes
1answer
99 views

Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
5
votes
1answer
171 views

Evaluate: $\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}}$

Evaluate the integral $$ P(\gamma)=\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}} $$ where $\gamma$ is a real number not equal to $0$ and has whatever ...
3
votes
2answers
55 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
1
vote
1answer
55 views

How would Cauchy calculate $\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$?

Please consider the following curve integral: $$I:=\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$$ where $$B_r(z_0):=\left\{z\in\mathbb{C}:|z-z_0|<r\right\}$$ Let $\gamma :[a,b]\to\Omega$ denote ...
5
votes
2answers
85 views

Riemann Zeta function Analytic continuation integral

Following Riemann paper about analytic continuation of Zeta Function: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf I can't understand the contour integral step: "If one now ...
0
votes
1answer
30 views

Evaluate $ \int_{C(0,5)} \frac{1}{i-\cos z}dz $.

How do I evaluate $$ \int_{C(0,5)} \frac{1}{i-\cos z}dz? $$ Do I have to find the poles first, and then use residue theorem, or find where the function is holomorphic and then integrate using ...
-1
votes
1answer
34 views

Quick question on Jordan's Lemma

The key equation in Jordan's Lemma is: $$I_\Gamma = \int_\Gamma e^{imz}f(z) dz \rightarrow 0$$ as $R \rightarrow \infty$. Why is $|\exp(imz)| = |\exp(-mR\sin\theta)|$?
3
votes
1answer
31 views

Quick question on poles

Consider this function for $0 < a < b$: $$f_{(z)} = \frac{z^4}{z^2(z-\frac{a}{b})(z-\frac{b}{a})}$$ This function has a pole of order $2$ at $z=0$, a pole of order 1 at $z=\frac{a}{b}$, but ...
0
votes
1answer
18 views

How do we find a path $\gamma$ with winding number $1$ and $2$ relative to points $1$ and $2$, respectively?

Let $\gamma :[a, b]\to\Omega\subseteq\mathbb{C}$ denote a parametric piecewise continuously differentiable path in $\Omega$ and $$\text{ind}_{\gamma}(z):=\frac{1}{2\pi ...
3
votes
1answer
62 views

Complex contour integral and partial fractions

I'm doing complex integration and I'm trying to evaluate: $$\int_C \frac{\cos{z}}{z^2 + 1} dz$$ Where $C$ is the clockwise boundary of a parallelogram with vertices $3i$, $2$, $-3i$, $-2$ (i.e. a ...
0
votes
0answers
27 views

Choice of Branch Cut in Contour Integral

Suppose we have to evaluate an integral: \begin{align*} \int_C f(z)\,dz \equiv\int_{C} \frac{e^{-iz}}{z^2-(a-bi)}dz\ ,\ a,b \in \mathbb{R}, a,b >0 \end{align*} where the contour $C$ is closed in ...
0
votes
3answers
57 views

Contour integrals using residues

The question I'm working on is the following: Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq ...
1
vote
1answer
99 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
3
votes
1answer
85 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
0
votes
1answer
26 views

Residue of $\frac{\cot{ax}}{x^2-b^2}$?

I am interested to find the residue of $$\frac{\cos{ax}}{(x^2-b^2)\sin{ax}}$$ at $x=b$. How would I go about doing this? I can see that the pole is second order, and so the formula $$\text{res} = ...
2
votes
0answers
64 views

Numerical integration of function with singularities

I am currently trying to solve a semi-infinite integral containing a set of singularities lying on the real axis numerically. The process I am using is breaking the integral into small steps $\Delta ...
8
votes
4answers
162 views

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis.

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the ...
0
votes
1answer
29 views

Complex Integral - exponential divided by a monomial

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $. EDIT: $\beta$ is a finite, real ...
1
vote
1answer
64 views

How is the multiplicity of a pole defined when square roots are involved?

A pole of multiplicity $m$ can be identified as the root of the denominator of a function like so: $$f(z) = \frac{g(z)}{(z-z_0)^m}$$ This also makes it easy to see that the $m$-th "order" residue ...
2
votes
1answer
35 views

Contour Integrals with Partial Fractions

Sorry, I'm new at posting here, so forgive me for any mistakes I make. I'm trying to evaluate the following using a contour integral. I don't know how to use the Residue stuff yet, so I basically ...
4
votes
2answers
133 views

Generalised Integral $I_n=\displaystyle \int_0^{\pi/2} \frac{x^n}{\sin ^n x} \ \mathrm{d}x, \quad n\in \mathbb{Z}^+.$

I have this integral, $$I_n=\displaystyle \int_0^{\pi/2} \frac{x^n}{\sin ^n x} \ \mathrm{d}x, \qquad n\in \mathbb{Z}^+.$$ We have the results $$ \begin{align} I_1 & = 2C, \\ I_2 &= \pi\log 2, ...
12
votes
3answers
291 views

Integral $\displaystyle \int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that ...
2
votes
1answer
51 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
135 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
55 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
36 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
32 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
53 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
42 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
39 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
21 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
34 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
96 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
109 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 ...
6
votes
0answers
212 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
226 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
82 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
394 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
54 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
64 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
94 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
91 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
82 views

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

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where $n$ is a non-negative integer. This problem is strongly ...
0
votes
0answers
48 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
73 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 $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
2
votes
1answer
157 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) ...