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9 votes
2 answers
2k views

Why does this theta function value yield such a good Riemann sum approximation?

Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e., $$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$ Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...
Timothy Chow's user avatar
  • 82.7k
6 votes
0 answers
150 views

Can this Casimir-effect integral be reduced to a special function?

This integral plays a central role in a physics problem (Casimir effect)${}^\ast$ $$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr]\,d\...
Carlo Beenakker's user avatar
0 votes
1 answer
138 views

Prove that the regularized incomplete beta function monotone with each of its parameter

Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...
Shiwen Yang's user avatar
2 votes
1 answer
111 views

Proof of the monotonicity of a regularized incomplete beta function

I want to prove the monotonicity of $I_r(nr, 2+(1-r)n)$ on $n$ but has no clues. The $I$ is the regularized incomplete beta function, defined as follows: $$I_r(nr, 2+(1-r)n)=\frac{\int_0^r x^{nr-1}(1-...
J H's user avatar
  • 25
1 vote
1 answer
79 views

PDF of the difference of two Beta Prime distribution

I am struggling to find the PDF of the difference of two Beta Prime distribution. Definition A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
NancyBoy's user avatar
  • 393
1 vote
0 answers
102 views

Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
Martin.s's user avatar
  • 224
5 votes
3 answers
845 views

Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

I uploaded this question here and here from my old account. Let $\psi^{(1)}$ be the trigamma function defined by \begin{equation} \tag{1} \psi_1(z) = -\int\limits_0^1 \frac{x^{z-1} \ln x}{1-x} \, dx. \...
Martin.s's user avatar
  • 224
6 votes
1 answer
406 views

Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE Certainly, I apologize for any oversight. Here's a more refined ...
Martin.s's user avatar
  • 224
1 vote
1 answer
265 views

Antiderivative of Meijer G-function

In the python sympy CAS framework one strategy to compute integrals is to transform the integration kernel to a MeijerG-function, obtain the corresponding antiderivative as MeijerG-function and ...
maliesen's user avatar
  • 284
2 votes
1 answer
133 views

How to calculate this integral of squared Tricomi hypergeometric function

How to solve this integral $$ \int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r $$ where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
tsukatsuki_sorano's user avatar
1 vote
1 answer
197 views

Derivation of indefinite integral involving hypergeometric function

I am doing a project on projectile motion and I ended up with this integral: $$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$ where $g, c,$ and $m$...
Leo McIntyre's user avatar
6 votes
1 answer
427 views

When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals, \begin{align} x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\ y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
Math_Y's user avatar
  • 287
3 votes
1 answer
499 views

Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?

The integral is $$\DeclareMathOperator{\dm}{d\!} \int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x. $$ Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
Guoqing's user avatar
  • 375
3 votes
1 answer
265 views

Integrating $\int_0^\infty \sqrt x e^{-4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$

Show that $$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$ $$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...
Zacky's user avatar
  • 215
2 votes
0 answers
252 views

Power series of the modified Bessel function of the second kind

I am looking for a power series representation of $$ \frac{1}{K_{\nu}(x)}, $$ where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer. I know that ...
esner1994's user avatar
0 votes
0 answers
81 views

Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
Ft insat's user avatar
2 votes
0 answers
70 views

How to extend this sum involving generalized harmonic numbers?

It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
Wolfgang's user avatar
  • 13.4k
8 votes
0 answers
296 views

Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?

I have never seen a real-analytic approach to evaluate integrals of the form below $$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
TheSimpliFire's user avatar
3 votes
0 answers
166 views

An inequality for integrals involving Laguerre polynomials

Let $k\ge n$ and $$A(k,n)=\frac{ \Gamma[1+k]}{n!\Gamma[1+k-n]^2}\int_0^\infty \frac{e^{-r}r^{k-n}}{L_n(-r)} dr$$ where $$L_n(-r) = \sum_{m=0}^n \frac{\Gamma(1+n)}{\Gamma(1+m)^2 \Gamma(1+n-m)}r^m$$ is ...
MathArt's user avatar
  • 333
1 vote
0 answers
35 views

How to relate this integration with the integral expansion of special functions?

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
Rekha K.'s user avatar
6 votes
2 answers
270 views

Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
Julian's user avatar
  • 623
1 vote
1 answer
276 views

Exponential decay bound on integral

I have an integral of the form $$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$ where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$ I would to get a nice ...
Guido Li's user avatar
0 votes
0 answers
103 views

Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral: $$\frac{1}{\pi}...
Eduardo's user avatar
0 votes
0 answers
76 views

Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$

How could one calculate the closed form solution of this integral: $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$ Here the integral is ...
NuKuYul's user avatar
  • 71
5 votes
2 answers
434 views

How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?

I got an interesting question. Consider this integral: $$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$ It is clear that ...
xiangsha's user avatar
6 votes
1 answer
436 views

Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$

Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers. \begin{equation} (\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2 ...
Joe's user avatar
  • 545
22 votes
1 answer
1k views

A multiple integral that seems related to the $\zeta$ function at even integers

I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
Joe's user avatar
  • 545
3 votes
1 answer
183 views

On integral representation of Whittaker $W$ functions

According to NIST, the integral representation of Whittaker $W$ functions $$ W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
Y.Okuyama's user avatar
  • 373
2 votes
1 answer
194 views

Generalized Selberg integral

I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions $$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
esner1994's user avatar
3 votes
0 answers
269 views

definite integral with incomplete gamma function and exponential

While working with electron density computations in quantum chemistry, I encountered the following improper integral: $$ I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
Igor's user avatar
  • 31
0 votes
0 answers
72 views

Integration of fractional function over Rice distribution

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx \end{equation}...
hichem hb's user avatar
  • 377
1 vote
0 answers
74 views

Complex integration related to finite temperature number density correlation function of 1d free fermion

I am looking for an explicit formula for this complex integral. $$\oint_C \frac{d z}{2 \pi i} \frac{z^{-(x+1)}}{1+e^{-\beta(z+1 / z-\mu)}},$$ where $x\in \mathbb{Z},\ \beta,\mu\in \mathbb{R}\ $. The ...
s hukahi's user avatar
11 votes
1 answer
566 views

Integral representation of product of two Whittaker functions

Does anyone know anything about the following formula involving special functions: $$\begin{multline*} W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
Y.Okuyama's user avatar
  • 373
2 votes
0 answers
136 views

Integral of Legendre's function

Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$ where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...
rihani's user avatar
  • 61
3 votes
1 answer
394 views

Closed form for the integral of a squared Legendre function

Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first ...
西島晃彦 a.k.a. Teru-san's user avatar
0 votes
0 answers
93 views

Closed form for a double integral over the first quadrant of the $L^p$ disk

Is there, by any chance, a closed form for the following integral $$ I_p=\iint_{Q_p}(x+y)\log(x+y)dxdy, $$ where $Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}$, $0<p\leq\infty$ ? ...
user111's user avatar
  • 4,034
2 votes
1 answer
840 views

Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where $L$ denotes the ...
Tom26's user avatar
  • 23
0 votes
1 answer
233 views

Logarithm of an integral involving generalized real binomial coefficients

I could not find a closed form for this integral although I think it should have been studied. What is a good approximation to $I$ in $$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
VS.'s user avatar
  • 1,826
0 votes
0 answers
136 views

A complex integration formula

I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
Knight Wang's user avatar
2 votes
0 answers
249 views

Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
user142929's user avatar
4 votes
1 answer
351 views

Asymptotic behaviour of function using Fox $H$-function representation

In equation (9) of this paper, it is claimed that the limiting behaviour $$ \int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
user121642's user avatar
0 votes
0 answers
42 views

Specify modified error function in form of error functions

How can we express $\mathrm{erf}(\frac{t-a}{m})$ as a sum of functions of the form $\mathrm{erf}(t)$? I am developing a fitting routine and I encounter this integral: $$\int_{0}^{x}\mathrm{erf}\left(\...
Shankar_Dutt's user avatar
1 vote
0 answers
146 views

Differential equation with Fresnel integral

We have $\frac{y'(x)}{\cos(x)}=C(x)$ and need to find y(x). Generally we should express $y(x)$ through $C(x)$ and elementary functions. I can only do it through $C(x)$ and $S(x)$, or through $\Phi(x)$....
Paul Ivanov's user avatar
3 votes
1 answer
223 views

Ratio of Selberg integral

I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
neverevernever's user avatar
1 vote
1 answer
632 views

Time ordered integral involving beta function:

Any help on unpacking integrals of the following type, would be helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
cheyne's user avatar
  • 1,611
6 votes
1 answer
560 views

Asymptotic Expansion of Bessel Function Integral

I have an integral: $$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$ and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...
djbinder's user avatar
  • 275
7 votes
3 answers
515 views

Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove: $$ \int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1. $$ Numerically it seems to hold true. So I have made some attempts to ...
Ramanasa's user avatar
  • 419
-1 votes
1 answer
149 views

How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$ Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
Math Learner 's user avatar
1 vote
2 answers
2k views

Integral of product of Gaussian pdf and cdf [closed]

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution. $\Phi(x) = \int_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution. How does one ...
Magic-A's user avatar
  • 43
0 votes
0 answers
272 views

Gaussian integral over logarithm of shifted error function

Suppose we have the following integral: $$ \int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi}}e^{\frac{-z^2}{2} } \log\left( \text{erf} \, a (z-b) +1 \right), \ \ \ \ a,b \in \mathbb{R} $$ Does a closed-...
Hipstpaka's user avatar
  • 355