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Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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4 votes
2 answers
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Reference for $\int_{-\infty}^{\infty}e^{a x^4+b x^3+cx^2}dx\;$?

In my research I encounter an integral of the form $$ \int_{-\infty}^{\infty}{\rm e}^{\large ax^{4}\ +\ bx^{3}\ +\ cx^{2}}\,{\rm d}x\qquad a < 0,\quad b, c \in \mathbb{R} $$ So the integral is ...
Sam Hilary's user avatar
0 votes
0 answers
25 views

On the determinacy of the Hamburger moment problem associated to generalized Laguerre polynomials

Let $\alpha>-1$. It is well-known that the measure $d\mu:=x^{\alpha}e^{-x}$ is the unique positive measure on $\mathbb{R}$ making generalized Laguerre polynomials $(L_n^{\alpha}(x))_n$ into an ...
user536450's user avatar
-1 votes
0 answers
26 views

Exact evaluation of a sum over product of Bessel functions [closed]

I am interested in the exact evaluation of the sum $\sum_{k=-\infty}^{\infty} \,k\, J_{m-k}(x) J_{n-k}(x) $. Here $k,m,n$ are integers and $J_p(x)$ is the Bessel function of the first kind. A similar ...
AAN's user avatar
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2 votes
2 answers
98 views

Asymptotics of modified Bessel function of second kind

Let denote $K_\nu$ the modified Bessel function of second kind of argument $\nu\in(0,\infty)$. It is kown that for $x\in\mathbb{R}$, we have: $$K_\nu(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}$$ as $x\to+\...
NancyBoy's user avatar
  • 494
2 votes
0 answers
24 views

Orthogonality of Whittaker functions

Is there a known orthogonality property of Whittaker functions $W_{\kappa,\mu}(iz)$ with respect to the first index as an integral over the argument? I am particularly interested in the case $\mu=0$ ...
Matt Majic's user avatar
1 vote
2 answers
109 views

Definite integral $\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx$? [closed]

For positive $a$ and $b$, consider the integral $$ I(a,b)=\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx. $$ Does $I(a,b)$ have a closed-form expression (far-fetched hope)? If not, does it ...
vrata's user avatar
  • 21
0 votes
1 answer
57 views

Could the product of two Bessel functions of the first kind be expressed in terms of infinite series $J_n(x)J_m(\alpha x)$, where $n,m\in\{0,2\}$?

It is well known that the square of the Bessel function of the first kind of order zero has the Maclaurin series expansion $$ J_{0}(x)^{2} = \frac{1}{\sqrt{\pi}}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\...
Siegfriedenberghofen's user avatar
0 votes
0 answers
53 views

how to use Gauss Multiplication Formula for Gamma function?

I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$ $$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$ but I didn't ...
Faoler's user avatar
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0 votes
0 answers
11 views

Question regarding terms with negative coefficients in multivariate Fox H function

I was studying a paper where an integral expression in terms of Fox H function of multiple variable were used. The definition of multivariate Fox H (extracted from appendix A-1 of Mathai-Saxena) is as ...
K.K.McDonald's user avatar
  • 3,263
3 votes
0 answers
81 views

What is the origin of the non-analytic behavior of the integral $\int_0^\infty \frac{t^3}{t^4+1} J_2(x t) dt$?

I came across integrals of the type $I(x) = \int_0^\infty \frac{t^3}{t^4+1} J_2(x t) dt$ From numerical integration, it seems that when $|x| \rightarrow 0$, the integral is reaching a limiting value ...
Archisman Panigrahi's user avatar
1 vote
0 answers
60 views

How to express the definite integral of this hyperelliptic integral as the definite integral of the elliptic integral

How to express the definite integral of this hyperelliptic integral as the definite integral of the elliptic integral? Using the substitution formula \begin{cases} p=e^{-\frac{2 i \pi }{5}} t^2+\frac{...
Eufisky's user avatar
  • 3,257
5 votes
1 answer
78 views

The $j$-invariant on the 'critical line' and its zeros

It is well-known that for the $j$-invariant, we have $j\left(\frac{1+\sqrt{-n}}{2}\right)\in\mathbb R$ whenever $n>0$. Moreover, $j\left(\frac{1+\sqrt{-n}}{2}\right)=0$ for $n=3$ at the cusp. In ...
Wolfgang's user avatar
  • 1,042
4 votes
0 answers
179 views

Definite integral involving K Bessel function and a square root

I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are $$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) ...
lewismcombes's user avatar
5 votes
0 answers
101 views

Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
Nightmare Integral's user avatar
1 vote
0 answers
31 views

Mellin transform of confluent Lauricella hypergeometric function

The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow $$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...
K.K.McDonald's user avatar
  • 3,263
1 vote
0 answers
33 views

Derivation of the associated Legendre Polynomials

I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of $$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$ Can someone point to a proper ...
Lukas Kretschmann's user avatar
4 votes
1 answer
95 views

Integral of $\exp(\langle x, a+ib\rangle)$ over hypersphere

I am looking to compute the following integral over $S_{n-1}$ the unit hypersphere in $\mathbb{R}^n$ \begin{equation} I(a,b) = \frac{1}{|S_{n-1}|}\int_{S_{n-1}}e^{\langle x,a+ib \rangle}dx \end{...
QLoop's user avatar
  • 43
3 votes
0 answers
58 views

What is the inverse Mellin transform of $\Gamma(1+i s)$?

Mathematica and WolframAlpha both indicate $$ \mathcal{M}_{s}^{-1}\left[\Gamma\left(1 + {\rm i}s\right)\right]\left(x\right) = -\,{\rm i} \operatorname{G}_{\,0,1}^{\,1,0}\,\left(x,{\rm i}\, \left\vert\...
Steven Clark's user avatar
  • 7,621
2 votes
1 answer
123 views

Inverse Laplace transform of product of three exponentials

Consider the Laplace Transform of the form $$ \operatorname{F}\left(s\right) = \frac{1}{\left(s - s_{1}\right)^{\large a_{1}} \left(s - s_{2}\right)^{\large a_{2}} \...
K.K.McDonald's user avatar
  • 3,263
0 votes
0 answers
19 views

Montonicity of incomplete beta function

Consider the function $f(x) = \log( \beta(x,\alpha, \beta) / x^{\alpha} )$ where $\beta(x,\alpha, \beta)$ is the incomplete beta function $$ \beta(x,\alpha, \beta) = \int_{t=0}^x t^{\alpha-1} (1-t)^{\...
David Harris's user avatar
  • 1,009
1 vote
0 answers
67 views

Laplace transform of special function

The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as $${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-...
K.K.McDonald's user avatar
  • 3,263
3 votes
0 answers
139 views

Is there a website that has all the special functions? [closed]

There are a lot of special functions, and I wonder if there is a website that collects all of them, similar to how the Encyclopedia of Triangle Centers collects information on triangle centers. ...
pie's user avatar
  • 6,416
1 vote
0 answers
12 views

Continuity of confluent hypergeometric function in terms of its parameters

The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as $${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{...
K.K.McDonald's user avatar
  • 3,263
1 vote
1 answer
96 views

Schrödinger equation quadratic potential

I am interested in the equation $y''(x) = (2a + x^{2}) y(x)$: In particular, I am interested in which values the parameter $a$ yield solutions which are bounded at infinity. I am aware that it is a ...
Andrew's user avatar
  • 58
5 votes
2 answers
182 views

How to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1) x^{3/10}} dx$

$$ \mbox{How to solve the following integral ?}:\quad \int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x $$ I think it cannot be solved using ...
stephan's user avatar
  • 341
12 votes
3 answers
542 views

Complex analysis or real analysis books that have these special functions.

I saw an integral question that involved the digamma function, which I know nothing about, and I want to learn more about it, its properties, and other functions like the polylogarithm function and ...
pie's user avatar
  • 6,416
2 votes
0 answers
32 views

Definition and Use of the Schett Polynomial in the Jacobi Taylor Series

I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
Kyler Rusin's user avatar
0 votes
1 answer
48 views

How to Evaluate this Sum

I am trying to evaluate this summation $$ \sum_{\ell = 0}^{k} \gamma^\ell \binom{2k + m}{\ell} $$ where $k$ and $m$ are non-negative integers and $\gamma$ is a real number. I tried using Mathematica ...
Debbie's user avatar
  • 854
2 votes
2 answers
70 views

Reference on asymptotic expansion of Dawson integral

The wiki and several posts claim that the Dawson integral has the following asymptotic expansion for large (real) $x$, $$ e^{-x^2}\int_0^x e^{y^2}dy \sim \frac{1}{2x}+\frac{1}{4x^3}+\frac{3}{8x^5}+\...
John's user avatar
  • 13.3k
0 votes
0 answers
24 views

Finding the function $g(t)$ given its Laplace transform $F(1/s)$

I am trying to find a function g(t) given that its Laplace transform is F(1/s), where F(s) is the Laplace transform of another function f(t). I know that if f(t) has the Laplace transform F(s), then f(...
Dr Potato's user avatar
  • 812
3 votes
0 answers
119 views

Gauss Hypergeometric Function Simplification

Recently I've been developing a strategy to handle a large class of Feynman integrals that uses some series expansions to simplify the angular integrals. This always leads to multiple sums over ...
y9QQ's user avatar
  • 89
0 votes
0 answers
28 views

Multi-variate incomplete beta functions

I am trying to generalize the following integral $$I_0=\int_{[0,1]^n}\text{d}\vec{x} \delta\left(\sum_{j=1}^{n}x_j-1\right)\prod_{i=1}^{n}x_i^{-p}=B_n(1-p,\dots,1-p)$$ where $B_n$ is the multi-variate ...
RKLS's user avatar
  • 81
1 vote
1 answer
68 views

How to find an expression for the $n$th partial derivatives of $1/r$?

From Pirani's lectures on General Relativity I got the following identity which he asks the reader to prove by induction which is easy $$\partial_{\alpha_1} \partial_{\alpha_2} ... \partial_{\alpha_n} ...
Sanjana's user avatar
  • 265
5 votes
1 answer
205 views

How to prove the result of the following integral? [duplicate]

How to prove that $$ \int _0^{\infty }\frac{K\left(\frac{1}{2}-\frac{1}{2 \sqrt{x+1}}\right)}{\sqrt[4]{x+1}}e^{-x}{\rm d}x = \frac{1}{2} \sqrt{e \pi } K_{1/4}\left(\frac{1}{2}\right) $$ where $K(x)$ ...
Jie Zhu's user avatar
  • 239
1 vote
0 answers
49 views

Is there a recursive identity for the derivative of the gamma function $\Gamma'(x)$ [closed]

For the normal gamma we have: $$\Gamma(x) = (x-1)\Gamma(x-1)$$ For the digamma we have: $$\Psi(x) = \Psi(x-1) + \frac{1}{x-1}$$ Is there something similar for $\Gamma'(x) = f(x-1)$?
terraregina's user avatar
0 votes
1 answer
44 views

Another general form of beta function evaluation

Is there a general formula for this integral, $$\int_0^1 \frac{x^a (1-x)^b}{(c+hx)^{a+b+2}}dx$$ I encountered an integral which broke into two smaller integrals and both of them were of the above ...
MathStackexchangeIsMarvellous's user avatar
1 vote
0 answers
121 views

How to evaluate the integral $\int_0^\pi e^{i(a\sin x + b\cos x)} dx$

We know that (from e.g. here) $$ \int_0^{2\pi}{\rm e}^{{\rm i}\left[a\sin\left(x\right) + b\cos\left(x\right)\right]}{\rm d}x = 2\pi\operatorname{J}_{0}\left(\sqrt{a^{2} + b^{2}}\right) $$ where $\...
userflux9674's user avatar
2 votes
0 answers
170 views

Is this multivalued inverse logarithmic integral $\operatorname{li}(x)<0$ power series valid?

$\DeclareMathOperator\li{li}$To derive the multivalued inverse of the logarithmic integral $y=\li(x)$$;y<0,x>1$, denoted $\li_-^{-1}(x)$ and for $0\le x<1$, denoted $\li_+^{-1}(x)$, we use: ...
Тyma Gaidash's user avatar
1 vote
0 answers
21 views

PDF of the difference of two Beta Prime distributions

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
  • 494
0 votes
0 answers
53 views

$ \lambda^{*}(n) $ minimal polynomials

I already asked a closely related question on MSE. Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here. Is there a way to calculate the ...
user967210's user avatar
2 votes
1 answer
77 views

Calculate Bell polynomial $B_{l,k}(x, x^2, x^3, \ldots,x^{l-k+1})$.

I am trying to show that $$\sum_{k=1}^{l}({-}1)^{k{-}1}(k{-}1)!B_{l,k}(x,x^{2},\ldots ,x^{{l-k+1}}) = 0$$ where $B_{l,k}(x_1,x_{2},\ldots ,x_{{l-k+1}})$ is the partial (or incomplete) Bell polynomial. ...
user3236841's user avatar
2 votes
2 answers
154 views

Solving an "impossible" integral, $\int \frac{dx}{a^{x^2}+b^{x^2}} $

First a saw this problem $$\int \frac{dx}{a^{x^2}+b^{x^2}} $$ where a,b are natural numbers This problem would be easier if $a=b$ then error function appears here I have tried with Wolfram alpha but ...
3e Ke 3m's user avatar
1 vote
0 answers
74 views

Proof of the Orthogonality of Hermite Polynomials

My question is regarding the proof of the orthogonality of Hermite polynomials. Actually, it's not quite the Hermite polynomials: $$ \psi_n(x) = [\dfrac{1}{\sqrt{n} 2^n n!}]^{\frac{1}{2}} e^{-\frac{x^...
Hooman Puyandeh's user avatar
17 votes
1 answer
377 views

Inverse function of the Exponential Integral $\mathrm{Ei^{-1}}(x)$

The Exponential integral is defined by $$ \mathrm{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm dt, $$ and has the following expansion $$ \mathrm{Ei}(x) = \log x + \gamma + \sum_{k=1}^\infty \frac{x^...
Nolord's user avatar
  • 1,661
0 votes
0 answers
24 views

Simplifying Meijer-G function with constant multiple argument

I have a function that can be written in terms of Meijer-G function as follow $$G_{1,2}^{2,1}\left(cz\left|\begin{smallmatrix}1-a \\ 0,1-b\end{smallmatrix}\right.\right)=\frac {1}{2\pi i} \int_L \...
K.K.McDonald's user avatar
  • 3,263
3 votes
0 answers
53 views

Find $\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$ algorithmically

It sometimes happens that $$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$ is algebraic for positive integers $m,n,a_k,b_k$. For example, $$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
Nomas2's user avatar
  • 667
2 votes
1 answer
88 views

Asymptotics of $\int_{0}^{1} \frac{\tan^{-1}(x^n)}{\sqrt{1 - x^n}} \, dx $

Working around this question, I tried to compute $$I_n=\int_{0}^{1} \frac{\tan^{-1}(x^n)}{\sqrt{1 - x^n}} \, dx $$ which I have been unable to express using Mathematica for general $n$. Some tedious ...
Claude Leibovici's user avatar
2 votes
2 answers
75 views

What is the correct argument definition for the complete elliptic integral?

For the complete elliptic integral of the first kind, we find on Wikipedia: $$ K(k) = \int_\limits{0}^{\pi/2} \hspace{-1ex} \frac{d\theta} {\sqrt{1 - k^2 \sin^2 \theta}} $$ But in Abramowitz and ...
Jos Bergervoet's user avatar
0 votes
2 answers
58 views

Finding an approximation for the complete elliptic integral of the first kind $K(k)$

Finding an approximation for the complete elliptic integral of the first kind $K(k)$ In this question the author find the following approximation to the integral: $$ I(k) := \int\limits_{0}^{\frac{\pi}...
Joako's user avatar
  • 1,362
14 votes
3 answers
341 views

Does anyone know what is this number?

Answering this question, @user64494 came with a very nice answer. My problem is that the front factor is HypergeometricPFQ[{1/2, 1/2, 1, 1}, {3/4, 5/4, 3/2}, -1] ...
Claude Leibovici's user avatar

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