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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12
votes
0answers
249 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
9
votes
0answers
321 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
37 views

Does such divergent integral assume the same values for any regularization?

Consider the integral: $$\int_0^\infty\sin(x)dx.\tag1$$ It's clearly divergent, but if we regularize it as $$\int_0^\infty\sin(x)e^{-x/a}dx=\frac{a^2}{a^2+1},\tag2$$ we can take the limit of ...
7
votes
0answers
150 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
213 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}{{\rm d}t \over t}\,{1 \over t - s - {\rm i}\epsilon}\, {1 \over \,\sqrt{\, 1 - a/t\,}\,}\, \ln\left(1 - ...
5
votes
0answers
167 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} ...
4
votes
0answers
69 views

How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ ...
4
votes
0answers
31 views

Integral representation of Bessel function K

Does someone have an idea how to connect the following function (appearing in the quantization of a real scalar field in a uniformely accelerated frame) : $$ K(x,y) = \int_{0}^{\infty} \frac{dt}{t} ...
4
votes
0answers
124 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
4
votes
0answers
44 views

Integral involving modified Bessel function of the second kind

I would like to calculate the closed-form expression for the following integral: $$ I = \int_{0}^{\infty} x^{M}\exp(-\frac{x}{a})K_{\nu}(b\sqrt{1+x})\mathrm{d}x,$$ where $M$, $a$, and $b$ are all ...
4
votes
0answers
111 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
4
votes
0answers
36 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
4
votes
0answers
59 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
4
votes
0answers
83 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
99 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
133 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
266 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
60 views

Looking for a rigorous treatment of improper multiple Riemann integrals

I'm studying undergraduate-level differential and integral calculus and have recently come across the topic of improper Riemann integrals. I'm familiar with the concept for single-variable functions, ...
3
votes
0answers
60 views

Can $\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$ be evaluated in closed form?

Can the integral $$\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$$ be evaluated in closed form using some known special functions? ($c \in \Bbb C$) This is taken from a question on ...
3
votes
0answers
27 views

Local behavior of a Fourier series and a intgral

So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is ...
3
votes
0answers
71 views

A hard integral from probability theory

I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables: $$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} ...
3
votes
0answers
61 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
3
votes
0answers
93 views

How can I show that the integral equals zero?

Problem: Show that $$\int_{0}^{\pi / 2} \ln\left(\tan x - \sqrt{2 \tan x} + 1\right)\,\mathrm{d}x = 0 $$ I'd like to use, if possible, only single-variable Calculus methods, and it does not include ...
3
votes
0answers
124 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
3
votes
0answers
48 views

integral over product of two bessel functions at discontinuity

The Weber-Schafheitlin integral $$ \int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(bt)}{ t^{\lambda}} $$ where $J_{\mu}(x)$'s are Bessel functions of the first kind, can have delta function singularities ...
3
votes
0answers
127 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
3
votes
0answers
124 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
3
votes
0answers
69 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
3
votes
0answers
373 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
113 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
3
votes
0answers
152 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 ...
3
votes
0answers
77 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
83 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
103 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
55 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
96 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 + ...
3
votes
0answers
77 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
127 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 ...
3
votes
0answers
161 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. $$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$ ...
3
votes
0answers
256 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
33 views

Are Riemann-integrals supposed to have a finite value?

Specifically, I am dealing with a task that implies that constant functions are not Riemann-integrable over $[0, \infty[$ unless $f=0$. Is that true? I didn't manage to find anything on that ...
2
votes
0answers
50 views

Asymptotic expansion of integral of e^(-t)/t^n

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}$$. I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now ...
2
votes
0answers
18 views

How to calculate this integral (with limit afterwards)?

I have to calculate this integral with limit: $$ G(m,n; E) = \lim_{\epsilon \rightarrow 0^+} \iint_{-\pi}^{+\pi} d k_x d k_y \frac{e^{i(k_x m + k_y n)}}{E+ i \epsilon + 2\cos k_x + 2\cos k_y } . $$ ...
2
votes
0answers
36 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
2
votes
0answers
45 views

Evaluating the integral $\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$

I'm searching for a way to evaluate the following integral: $$\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$$ where $J_v(x)$ are the Bessel-functions, and $v \in \mathbb{N}, (a,\beta) \in ...
2
votes
0answers
59 views

Value of improper Integral

I need help in finding the value of the integral $$\displaystyle \int_0^\infty \left(\frac{x^2}{1+x}\right)^{n-1}e^{-tx}dx,$$ where $n$ is a positive integer and $t$ is a positive real number.
2
votes
0answers
54 views

improper integrals in q-calculus

In quantum calculus is this equality possible for improper integrals? $\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
2
votes
0answers
60 views

Is there a closed-form expression for this trigonometric Cauchy Principal Value-type integral?

Consider the following definite integral, $I(n; \theta)$. $$ I(n; \theta) = \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N $$ When $0 < \theta < \pi ...
2
votes
0answers
24 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
2
votes
0answers
50 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...