Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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3
votes
0answers
114 views

Limit with the Appell F1 function

While attempting to solve this problem I ran into a nasty limit. Mathematica claims that the indefinite integral ...
0
votes
0answers
14 views

Wigner 3j for Interchanged $m$

I am given with two wigner j coefficients $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ m& 0& -m\end{pmatrix},$$ and $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ -m& 0& ...
8
votes
1answer
236 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
0
votes
0answers
20 views

Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
2
votes
1answer
42 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
0
votes
0answers
20 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
0
votes
0answers
30 views

The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
1
vote
0answers
31 views

Integration on complex spheres and Gamma function

I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), ...
0
votes
0answers
34 views

Integral in terms of special function

I'm trying to express the integral given below in terms of special functions. $$\int_{1}^{\infty}\left(-1\right)^{n}x^{-1-a}\left[\ln\left(e^{\left(x-1\right)/b}-1\right)\right]^{n}\mathrm{d}x$$ I ...
16
votes
2answers
519 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
3
votes
0answers
64 views

What is the series expansion of reciprocal of theta function $\frac{1}{\theta(z;q)}$?

"The" theta function is an ambiguous concept, but one definition I have found is: $$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} ...
0
votes
1answer
43 views

Does this limit involving the Dirichlet eta function and the Riemann zeta function make sense?

Let $p_n$ the sequence of prime numbers (and you will consider below, too, the sequence $\frac{1}{n}$ with $n>1$). And if it isn't wrong for $0<\Re s<1$ the known equation between Dirichlet ...
2
votes
1answer
42 views

Is the Lambert W function multivalued everywhere?

Is the Lambert W function multivalued everywhere except at $x=0$? It is obvious that $W(0)=0\implies 0=0e^0$ because $e^u\ne0$, therefore, it is the coefficient that determines such, and the only ...
2
votes
1answer
102 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- ...
1
vote
1answer
55 views

Express $\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt$ in terms of $x$ and $n$

Please help me to express $$\int_{\sin nx}^{\sin(n+1)x}\sin t^2\,dt$$ in terms of $x$ and $n$. If it is not possible please help to establish bounds on the integral again in terms of $x$ and $n$. The ...
0
votes
0answers
45 views

How to prove the following function is independent of z?

I series expanded the following expression in Mathematica and the result is independent of z: $$(1-z)^{-a }\left(\, _2F_1(1,-a ;1-a ;1-z)+\, _2F_1\left(1,a ;a+1;\frac{1}{1-z}\right)-1\right)-(1-z)^{a ...
1
vote
1answer
53 views

Relation between hypergeometric functions?

Is there any relations between the following hypergeometric functions? $$\ _2F_1(1,-a,1-a,\frac{1}{1-z})$$ $$\ _2F_1(1,-a,1-a,{1-z})$$ $$\ _2F_1(1,a,1+a,\frac{1}{1-z})$$ $$\ _2F_1(1,a,1+a,{1-z})$$
22
votes
1answer
292 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
0
votes
1answer
36 views

How to prove this HyperGeometric function identity?

After using FullSimplify in Mathematica, I got the left hand side of the following equation: $$(a-1)(z-1)\ _2F_1(1,1,1-a,\frac{1}{z})+a\ _2F_1(1,1,2-a,\frac{1}{z})-az+z=0$$ I series expanded it and ...
3
votes
2answers
145 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
2
votes
2answers
421 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
1
vote
1answer
33 views

Show that $f(qz) = qz^2 f(z)$ where $f(z) = [\theta(z) \theta(1/z)]^{-1}$

Let $\theta(z) = (z;q)(q/z; q)$ where $(q;z) = \prod_{i=0}^\infty (1 - zq^i) $. Then let $f(z)$ be defined by: $$ f(z) = \frac{1}{\theta(z) \theta(1/z)} $$ Show that $\boxed{\color{#0033FF}{f(qz) = ...
2
votes
1answer
62 views

Which theorems in the Gamma Function are important? [closed]

I'm interested in the Special Functions especially the Gamma Function. I decided to write a Bachelor Thesis about it but I do not know what kind of theorem(s) in the Gamma Function that is (are) very ...
31
votes
1answer
497 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
0
votes
0answers
52 views

Express $\cos^2\theta\cos\phi\sin\phi$ in Spherical Harmonics

I am looking for a form of $$\cos^2\theta\cos\phi\sin\phi=\sum_{lm}c_{lm}Y_l^{m}(\theta,\phi),$$ where $Y_{lm}$ is the spherical harmonics. The idea I believe would be to find ...
1
vote
1answer
49 views

Monomials in terms of Legendre polynomials

Is there a closed-form expression for a monomial $x^m$ in terms of a sum of Legendre polynomials $P_n(x)$? $$ x^m = \sum_n a_n P_n(x) $$ How can I determine the coefficients $a_n$ in general? ...
9
votes
3answers
829 views

Integral with Bessel functions of the First Kind.

I'd like to solve the following integral: $I = \int_0^\infty J_0(at) J_1(bt) e^{-t} dt\ $ where $J_n$ is an $n^{th}$ order Bessel Function of the First Kind and $a$ and $b$ are both positive real ...
10
votes
1answer
182 views

Elliptic functions as inverses of Elliptic integrals

Let us begin with some (standard, I think) definitions. Def: An elliptic function is a doubly periodic meromorphic function on $\mathbb{C}$. Def: An elliptic integral is an integral of the form ...
0
votes
0answers
14 views

Legendre's Associated Functions

$$Legendre's \ Associated\ Functions :$$ I.N.Sneddon-Special Functions : Page (74) $$\left \{ \left ( 1-\mu ^{2} \right )\frac{\mathrm{d^{2}} }{\mathrm{d} \mu ^{2}}-2\mu \frac{\mathrm{d} ...
6
votes
1answer
106 views

Solution to $xe^{e^x}$

The problem $xe^{e^x}=e$ came up another day and I wondered if it were solvable. My attempt was the following substitution,$$x=W(u)$$$$W(u)e^{e^{W(u)}}=e$$Where I used a Lambert W identity to get ...
1
vote
1answer
52 views

Copulas: Grounded or increasing functions.

For a function $H(x,y)$ to be a copula, it has to be increasing in $x$ and in $y$. But, instead of this condition, other authors say that the function has to be grounded. Are these properties ...
1
vote
0answers
28 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
1
vote
1answer
35 views

residue equation for the denominator in a Padé approximant for $e^{-x}$

I had success in computing the roots numerically for the Bessel polynomial $\theta_n(x) = x^ny_n(1/x)=\sum\limits_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\frac{x^{n-k}}{2^k}$ by using this residue equation I ...
0
votes
1answer
43 views

An identity about Bessel functions

How can I prove $\frac { 2n}{\rho}J_n (\rho)=J_{n-1}(\rho)+J_{n+1}(\rho)$ ? When $J_n$ is n'th order Bessel function. I tried a lot, but I don't know how to construct $"n"$ in the LHS. Is ...
0
votes
1answer
62 views

Integrate this Spherical Harmonics Function [closed]

I am interested in the following integral $$\int_0^{2\pi}\int_0^\pi\mathop{\mathrm{d}\theta}\mathop{\mathrm{d}\phi} \sin\theta Y_l^{m*}(\theta,\phi)Y_{l'}^{m'}(\theta, \phi)\cos^2\theta\cos^2\phi,$$ ...
1
vote
0answers
55 views

Inverse function of hypergeometric function, e.g., ${}_{2}F_{1}(1,1;1.2;x)$

I want to know whether it is able to express the inverse function of hypergeometric function using some special function. For instance, the Gauss hypergeometric function ...
1
vote
1answer
66 views

Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$

While doing research in electrical engineering, I derived the following integral representation of the Bessel function of the first kind: ...
0
votes
0answers
24 views

Does this integral of Appell F_1 converge?

I'm interested in whether or not integrals of the form $$\int_{0}^{1}\mu^{\alpha}F_{1}\left(\frac{\alpha}{2};1,-1;\frac{\alpha+2}{2};\mu^{2},-\beta\mu^{2}\right)\mathrm{d}\mu$$ converge, and if so ...
4
votes
2answers
79 views

Lipshitz Integral for $a=0$

I knew that this, $$\displaystyle{\int_0^\infty e^{-ax}J_0(bx)dx=\frac{1}{\sqrt{a^2+b^2}}},$$ holds for $a>0$ but, in an exercise from Arfken, it said that this holds for $a\geq0$. How can I prove ...
1
vote
0answers
36 views

Asymptotic of $ _1F_1(a;b;z)$

How it can be shown that $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(a)}\, e^{z} \, z^{a-b}\, [1+ O(\mid z\mid^{-1})]; \quad (\Re(z)>0)$$ or $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(b-a)}\, ...
2
votes
2answers
38 views

Find out $n$-th term of monotonic functions increasing and decreasing

I have a series whose max and min values are defined. the values in the series have an increase monotonically by $x\%$ and decrease once the maximum is reached. For example, this series has a min ...
1
vote
0answers
28 views

Proof of Hypergeometric Contiguous relation

I want to prove the following recursive relation: $$c(c+1)_2F_1(a,b;c;z)=c(c-a+1)_2F_1(a,b+1;c+2;z)+a[c-(c-b)z]_2F_1(a+1,b+1;c+2;z)$$ I tried using both the series ...
7
votes
1answer
132 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
7
votes
1answer
233 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
1
vote
1answer
38 views

Confusing solution to the limit of an implicit function?

$$\frac{8}{3}=\frac{\log{x}}{\log{y}}-\frac{\log{y}}{\log{x}}$$ When I graphed this implicit function on desmos (https://www.desmos.com/) it appeared as if there were two solutions as $x\to{0}$ from ...
3
votes
2answers
1k views

Second order linear ODE, self adjoint (Sturm-Liouville) form. Orthogonality of solutions - confused about the weight factor.

If I have an ODE of the form $$a(x)y''+b(x)y'+c(x)y= \lambda y$$ Such that $b=a'$, then it is equivalent to: $$(a(x)y')'+c(x)y= \lambda y$$ So the solutions corresponding to two different ...
4
votes
2answers
102 views

Finding a Particular Solution for $\frac{d^2R}{dr^2}+\frac{1}{r} \frac{dR}{dr}+\alpha^2R=J_0(\alpha r)$

Motivation I have the following non-homogeneous Bessel differential equation $$\frac{d^2R}{dr^2}+\frac{1}{r} \frac{dR}{dr}+\alpha^2R=J_0(\alpha r)$$ I want to find the general solution for this ...
7
votes
1answer
149 views

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page ...
2
votes
0answers
62 views

Norm of the inverse of a map $\ell^2\to\ell^2$

Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in \mathbb Z}, u_i\in \mathbb R:\sum_{i\in \mathbb Z}u^2_i<+\infty\}; \beta>0$. How to compute $||A^{-1}||$ or estimate it? ...
4
votes
2answers
91 views

Calculate $‎\lim‎_{ ‎r\rightarrow ‎\infty‎}‎‎\frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)}‎‎$

I need to calculate limit $$‎\lim‎_{ ‎r\rightarrow ‎\infty‎}‎‎\frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)}‎‎$$ where $0<\alpha <1$ and $\Gamma(.)$ is Gamma function. with thanks in advance.