2
votes
1answer
37 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
4
votes
0answers
51 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
7
votes
1answer
132 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
2
votes
3answers
48 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
1
vote
3answers
48 views

integrating $\int_{\gamma}e^zdz$ with $\gamma$ is the arc on the unit circle that unites one with i

I am stuck integrating $$\int_{\gamma}e^zdz$$ with $\gamma$ is the arc on the unit circle that unites one with i. I tried this : The integrand $\mathrm{e}^z$ is holomorphic for $\vert z \vert \le ...
9
votes
0answers
147 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
3answers
108 views

Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$ I did integrate it by parts, by ...
2
votes
2answers
159 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
3
votes
2answers
67 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
2
votes
2answers
146 views

Gaussian integral with offset, and other cases

Consider the Gaussian Integral $$ \int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}$$ Numerically, it seems that for any arbitrary imaginary offset, ki, $$\int_{ki-\infty}^{ki+\infty} e^{-x^2} \ dx ...
0
votes
0answers
26 views

How to calculate $\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$ using Cauchy's integral theorem? [duplicate]

I need a hint. Where do I start if I want to calculate $$\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$$ using Cauchy's integral theorem?
3
votes
0answers
117 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
1
vote
0answers
48 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
0
votes
1answer
22 views

The integral along a circle of the inverse linear function is zero

Assume ${\rm C}$ is a circle and $a,b$ are distinct points in the interior of ${\rm C}$. How can we see that the complex integral $$ \frac{1}{b - a} \int_{\rm C}\left(\frac{1}{z - a} - \frac{1}{z - ...
8
votes
1answer
255 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
0
votes
2answers
34 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
2
votes
3answers
67 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
0
votes
0answers
86 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
2
votes
0answers
56 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
1
vote
0answers
31 views

Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
0
votes
0answers
18 views

Divergence of Euler integral for non-positive arguments

Why is it necessary that $\operatorname{Re}(x),\operatorname{Re}(y) > 0$ for the Beta-function $$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$$ I suppose it is because the integral diverges when ...
2
votes
0answers
53 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
0
votes
0answers
29 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
2
votes
1answer
32 views

Lusin's area integral

I was reading "Steven G. Krantz - Handbook of Complex Variables" and came around a complex surface integral called "Lusin's area integral": If $\Omega \subseteq \mathbb{C}$ is a domain and $\varphi: ...
0
votes
0answers
58 views

Integral $\int^\infty_{-\infty}\int^\infty_{-\infty}(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1)^{-a_1-1}(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1)^{-a_2-1}dxdy$

Under $x_i,y_i\in\mathbb R$, $s_i>0$ and $a_i>0$ for $i=1,2$, is there any good function to express the following integral? $$\int^\infty_{-\infty}\int^\infty_{-\infty} ...
0
votes
2answers
77 views

Calculate: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$

I would like to calculate this integral: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$ I calculated : $\displaystyle ...
4
votes
2answers
151 views

Integration of exponential and square root function

I need to solve this $$\int_{-\pi}^{\pi} \frac{e^{ixn}}{\sqrt{x^2+a^2}}\,dx,$$ where $i^2=-1$ and $a$ is a constant.
1
vote
1answer
36 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
3
votes
3answers
166 views

Integrate $1/(x^5+1) $from $0$ to $\infty$?

How can I calculate the integral $\displaystyle{\int_{0}^{\infty} \frac{1}{x^5+1} dx}$?
3
votes
1answer
99 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
0
votes
0answers
21 views

curvilinear integral: $\oint_{C^{+}}F dx $ and $F(x_1,x_2,x_3) = [\frac{-x_2}{x_1^2+x_2^2}, \frac{x_1}{x_1^2+x_2^2}, 0]$

$ C=(\cos t \cos\sin(nt), \sin t \cos \sin(nt), \sin \sin(nt)): t \in [0,2\pi]$ a) Find $$\oint_{C^{+}}F dx $$ and $F(x_1,x_2,x_3) = \left[\dfrac{-x_2}{x_1^2+x_2^2}, \dfrac{x_1}{x_1^2+x_2^2}, ...
2
votes
2answers
53 views

Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^{\infty}\frac{e^{itx}}{\pi(1+x^2)}dx$ using residue theorem. Is there any other way to calculate this integral (for someone who ...
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
3
votes
2answers
95 views

Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
2
votes
1answer
22 views

weighted integral in convex hull

Working on an integral $$ J=\frac1{2\pi} \int_0^{2\pi} w(t) g(e^{it}) dt $$ where $\frac1{2\pi} \int_0^{2\pi} w(t) dt=1$ ; $w(t)$ is non-negative continuous ...
2
votes
1answer
44 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
1
vote
0answers
22 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
4
votes
1answer
141 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
1
vote
1answer
101 views

Complex Integral Question

I'm trying to evaluate the following integral, in preparation for my exam tomorrow; $$\int_{0}^{\infty} \frac{\cos(2x) - 1}{x^2} dx$$ However, I'm having a lot of issues with it. I was initially ...
0
votes
0answers
16 views

analyticity of an integral

Studying $$f(z)=\int_0^1 g(z,x)\ dx $$ where $g(z,x)$ is analytic in the open unit disk $D$ for all $x\in [0,1]$ and continuous for $|z| <1$ and $0\leq x \leq1$.Now, $$\lim_{n\to\infty} [1/n ...
0
votes
0answers
31 views

Line Integrals in complex plane

I recently learned about line integral in the complex plane in my Complex Analysis class but I'm struggling a bit with some exercises, namely: Calculate $\oint_c f(z) dz$, where: a) $f(z)= ...
1
vote
0answers
52 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
2
votes
1answer
51 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
1
vote
1answer
81 views

Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths [duplicate]

How can I calculate $$\int_0^\infty\frac{\sin x}xdx$$ by integration of a suitable function along the following paths: where $R$ and $\varepsilon$ are the radius of the shown outer and inner ...
2
votes
0answers
47 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
2
votes
1answer
54 views

Compute an integral with residue theorem

Using residue theorem, compute the following integral: $$ \int_{0}^{2\pi}\frac{\left( 1+2\cos t\right) ^{n}\cos\left( nt\right) }{5+4\cos t}\operatorname*{dt}. $$ Or a source with a solution.
1
vote
1answer
54 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
0
votes
1answer
24 views

the area of the image under a specific holomorphic function of the unit disk

Let $f(z)=z^3+\frac{z^2}{2}$. Let $D$ be the unit disk in $\mathbb{C}$. How to compute $$ Area(f(D))? $$ In the case that $f:D\to \mathbb{C}$ is injective, \begin{align*} Area(f(D))&= \int_D ...
1
vote
1answer
17 views

ML Inequality and Complex Integrals

I have a question about the ML-Inequalty and how it applies on complex integrals. Getting this page from Wikipedia http://en.wikipedia.org/wiki/Estimation_lemma the denominator in the f(z) is $(z² + ...
0
votes
1answer
21 views

Real part of integral over holomorphic 1-form is zero implies the one-form is zero

Suppose we are working on a Riemann Surface $X$, assume of genus g $\geq$ 1. Let $\omega$ be a holomorphic 1-form on $X$, with $$ \textrm{Re} \int_\gamma \omega = 0$$ for every closed contour ...