Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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24
votes
0answers
408 views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
11
votes
0answers
310 views

Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$

The Cauchy's Beta Integral is given by $$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ I would like to know how it is proved.
10
votes
0answers
398 views

Closed form for this integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}$

How would you evaluate this integral? \begin{equation}\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}\end{equation} It reminds me of the form of a modified Bessel function of the ...
8
votes
0answers
81 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
8
votes
0answers
267 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda ...
7
votes
0answers
165 views

Evaluating $\int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2}$ using real methods.

By reading a german wikipedia (see here) about integrals, i stumpled upon this entry 27 1.5 $$ \color{black}{ \int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2} ...
6
votes
0answers
140 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...
5
votes
0answers
178 views

A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $

Can anyone give any hints on how to rewrite this in terms of dilogarithms? $$\int_0^\infty ...
4
votes
0answers
57 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
4
votes
0answers
78 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
4
votes
0answers
126 views

Taking an integral of e^(another integral) with respect to the limit of integration of (another integral)

How would I go around integrating $$\int_0^\infty \left( \exp{\left(-\int_{k_0}^{k_0+t} (k_1 + k_2s)(k_3+k_4s)^{s} ds \right) } \right) dt \text{,}$$ where $k_i$ are constants? Is it solvable ...
4
votes
0answers
179 views

How do I express $\int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz$ in terms of named functions?

Recently I derived an expression for a particular probability density function. The expression contains the integral $$ f(t,v,a) = \int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz = 2a ...
4
votes
0answers
243 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
3
votes
0answers
270 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
3
votes
0answers
65 views

Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
3
votes
0answers
54 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
3
votes
0answers
77 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
3
votes
0answers
41 views

Convergence of a integral

The question is: exists a natural number $n \geq 2$such that $$ \displaystyle\int_{0}^{+ \infty} \displaystyle\frac{\ln r}{(1 + r^2)^{n}} r dr< \infty ?$$ I am trying to do this : i know that ...
3
votes
0answers
38 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
3
votes
0answers
56 views

What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?

I want to compute the following integral: $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$ with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$. ...
3
votes
0answers
143 views

Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
0answers
221 views

2 dimensional Fourier transform integral

I'm trying to calculate the two dimensional Fourier integral $$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$ with $\vec{R}=(x,y)$. Switching to ...
2
votes
0answers
23 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
2
votes
0answers
32 views

Which substitution do I use?

For $x>0$, let $f(x)=\int_0^\infty e^{-t-x^2/t} t^{-1/2}dt$. a) show that $f(x)=x\int_0^\infty e^{-t-x^2/t} t^{-3/2}dt$ via an adequate substitution. b) Calculate $f'(x)$ and show that ...
2
votes
0answers
142 views

on the convergence of a certain integral

If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor ...
2
votes
0answers
95 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
2
votes
0answers
112 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
2
votes
0answers
41 views

Complicated Inequality involving Improper Integrals

Let $\ell>k>1$, $r=\frac{\ell}{k}-1$ and $y'$ be positive and $L^k$, then $$\int_{0}^{\infty}\frac{y^\ell}{x^{\ell-r}}dx<K \left( \int_{0}^{\infty} y'^k dx\right)^{\frac{\ell}{k}} $$ ...
2
votes
0answers
55 views

Fubini theorem for improper Riemann integral

Is there a version of Fubini's theorem for improper Riemann integrals? Here's an example of what such a version might look like. If $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is bounded and non-negative ...
2
votes
0answers
53 views

Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
2
votes
0answers
45 views

Numerical Methods for estimating divergence over an improper integral

Problem given a function $f(x)$, defined on $[ \epsilon, \infty )$. Is there a way to "numerically estimate" whether the integral of the function diverges over the domain $[ \epsilon, \infty )$? ...
2
votes
0answers
29 views

Integral to be computed

I am interested in computing $$J(a,b):=\int_{\mathcal{I}(a,b)}\frac{dx \ dy\ dz}{|ax^3+ay^2+bz^3|^{2/3}}, $$ where $a,b$ are natural numbers and $$\mathcal{I}(a,b):=(0,1]^3\cap\{x,y,z \in ...
2
votes
0answers
62 views

Frullani version of the classic $\log \left( 1 + 2\alpha \cos px + \alpha^2\right)$ integral.

I read in a paper about Frullani integrals the following claim $$ \begin{align*} I & :=\int_0^\infty \frac{1}{x}\log\left(\frac{1 + 2\alpha \cos px + \alpha^2}{1 + 2\alpha \cos qx + ...
2
votes
0answers
109 views

Integral involving bessel functions, exponential and ratio of polynomials

I need to solve this integral: $$\int_0^{+\infty}\quad\frac{k}{k^2-\alpha^2}\,J_1(2\pi R k)J_1(ak)\,\exp(-4\pi^2\omega^2k^2)\, dk$$ where $\omega,\,R,\,a>0$ and $J_1(a\alpha)=0$. Thanks!
2
votes
0answers
53 views

Solve the special integral

I want to solve a integral which contains a shift version $$\int^{\infty}_{c}N [(1-e^{-1/t})]^{N-1} \frac{-1}{(t-c)^2}e^{-1/(t-c)}dt$$ This kind of integral has the form of normal integral $$ \int ...
2
votes
0answers
93 views

computing a difficult integral using software

I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
2
votes
0answers
116 views

Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Being inspired by this post, I've wondered if the infinite series below may be expressed as an intregral. I'm very curious about that. $$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$ ...
2
votes
0answers
98 views

Analyzing the convergence of an improper integral

I have to analyze the convergence of $$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$ I've rewritten the integral as $$ \int _{0}^{+\infty} \frac{\cos \left( x\right) ...
2
votes
0answers
172 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
2
votes
0answers
127 views

Convergence of Riemann-like sums

I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that: \begin{equation} \sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+. ...
1
vote
0answers
29 views

convergence (absolutely) of an improper integral

$$\int_{-\infty}^\infty\frac{\sin(\sin x)}{1+\log(\lfloor|x|\rfloor! + 2)} dx$$ I need to check if this integral is absolutely convergent... I've shown it's convergent (not absolutely), according ...
1
vote
0answers
18 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
1
vote
0answers
68 views

What is the solution of the integral (product of two standard normal CDFs)?

I need to compute this kind of integral: where $b>0,d>0,a,c$ and $e$ are known constants and $\Phi$ is the CDF of the standard Normal distribution.
1
vote
0answers
78 views

solution of another definite integral

Does the following integral converge or not? \begin{align} && \sum_{k=0}^{\infty} (-\varphi)^k \binom{\frac1\varphi+k}{k}\int_{-\infty}^\infty\beta x^n e^{-\beta x(k+1)}dx&& ...
1
vote
0answers
37 views

solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
1
vote
0answers
55 views

How to integrate the following function?

Let's have the integral $$ I(\mathbf r, \omega) = \int \limits_{-\infty}^{\infty} e^{i(\mathbf k \cdot \mathbf r )}\frac{\sin(\omega \sqrt{\kappa^2 + k^2})}{\sqrt{\kappa^{2} + ...
1
vote
0answers
39 views

Example of continuous positive function without limit whose improper integral is convergent

I would like an example of a function that is continuous and positive and has the following properties: $$\int_a^{\infty}f(x) dx $$ is convergent and $$\lim_{x \to \infty} f(x) \not = 0$$ (I think the ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
1
vote
0answers
30 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
1
vote
0answers
53 views

Integral $\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx$

Does the following integral admit a closed-form expression? $$\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx \;\; , \;\; 0 \leq \alpha \leq ...