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

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6
votes
1answer
99 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
39 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
27 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
31 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
40 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
54 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
30 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
36 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: ...
43
votes
7answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

The question is in the title. To rephrase, does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if ...
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
74 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
33 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
35 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
25 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
118 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
227 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
31 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
98 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 ...
8
votes
1answer
138 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
57 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
89 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.
1
vote
1answer
35 views

Is there a name for the linear maps $u_i: E_i \to \prod_k E_k$ defined by $u_i(t) = (0,…,0,t,0,…,0)$?

Let $E_1,...,E_n$ be vector spaces. We know that a function $p_i: \prod_k E_k \to E_i$, $p_i(x_1,...,x_n) = x_i$ is called a projection function. I often have to use the function $u_i: E_i \to ...
0
votes
0answers
26 views

Gamma function converges to zero

I want to show that for $x \in [1,2]$ the Gamma function $\Gamma(x+iy)$ converges uniformly to zero as $y \rightarrow \pm \infty.$ Unfortunately I have not found a suitable representation of the Gamma ...
4
votes
2answers
80 views

Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$

I encountered the following integral in a physical problem $$I=\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$$ where $\text{J}_0$ is the Bessel function of first kind of order $0$ ...
2
votes
2answers
60 views

explicit formula for $ _2F_2(1,1;2;2;z) $

Is it an explicit formula for $$ _2F_2(1,1;2;2;z) ,$$ where $$_2F_2(a,b;c;d;z)=\sum_{n\geq 0}\frac{(a)_n(b)_n}{(c)_n(d)_n n!}z^n .$$ thanks you in advance
5
votes
1answer
49 views

Building a non decreasing function from any other function

What can be said of this function? Does it have a name? Would it be possible to build an equation for it? Is it implemented in any mathematical software? $$f_g(x)= \max \{g(y) \mid y\in [0,x]\}$$ I ...
0
votes
0answers
45 views

Solving for $x$ in $(y-x)\ln\frac{x}{y} = a$

I have the expression $$(y-x)\ln\frac{x}{y} = a,$$ and I want to express $x$ in terms on $y$ and $a$. I know that in this kind of problem, the Lambert function $W$ is likely to show-up, but that ...
2
votes
2answers
831 views

Spherical Bessel Zeros

I was wondering if there is a known closed form solution for the zeros of the spherical Bessel functions. While doing a quantum assignment, I came across them as a solution for the spherical infinite ...
7
votes
0answers
156 views

Polylogarithm ladders for the tribonacci and n-nacci constants

While reading about polylogarithms, I came across the nice polylogarithm ladder, $$6\operatorname{Li}_2(x^{-1})-3\operatorname{Li}_2(x^{-2})-4\operatorname{Li}_2(x^{-3})+\operatorname{Li}_2(x^{-6}) = ...
1
vote
0answers
31 views

Exact solution of an simple Meijer-G function

I am trying to simplify the following Meijer-G funtion \begin{equation} G^{2,2}_{2,2}\Bigl({}^{0,\, 1-m}_{0,\,0} |x \Bigr) \end{equation} But the Matlab(MuPAD) and WolframAlpha give me different ...
2
votes
0answers
36 views

Finding a relation between hypergeometric functions $_2F_1$

I would write the following Gauss hypergeometric function $$ _2F_1 \left(a,b; c-n; x\right) $$ in terms of $$ _2F_1 \left(a,b; c; x\right) $$ Where $a,b,c\in \mathbb C , x\in \mathbb R$ and $n\in ...
7
votes
1answer
94 views

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions. However, it does not explain why this is the case. An elementary function is "a ...
0
votes
1answer
37 views

Express the indefinite integral $\int e^{-x^2}dx$ using function $\Phi(x)$. [closed]

Express the indefinite integral $\int e^{-x^2}dx$ using function $\Phi(x)$. $\Phi(x)$ is the following special function: $$\Phi(x) = \frac12 +\frac{1}{\sqrt{2\pi}}\int_0^x e^{-t^2/2}\,dt$$
1
vote
1answer
49 views

Proof of $\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$

I encountered the following limit while doing calculation $$\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$$ which is equivalent to $$\lim_{x \to \infty }e^{-x}\sum_{n=1}^{\infty}\frac{x^n}{n·n!}=0$$ and ...
52
votes
0answers
585 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
3
votes
0answers
22 views

integral form of special function

Do you have any idea to present integral form of this function? $f(x)=\frac{1}{x^2}+\frac{1}{x}(\psi(x)-2\ln x-2)+2(1+\ln x)\psi^{'}(x)+x\ln x\psi^{''}(x).$ Where $\psi^{(n)}(x)$ is polygamma ...
0
votes
1answer
24 views

Finding the domain and Range of a piece wise Function,

Can someone explain to me how to find the domain and range of a piece wise function using this example? Thanks
2
votes
1answer
48 views

Does a limit to this Hypergeometric Function Exist Analytically?

I am interested in evaluating limit $$\lim_{x\rightarrow\pi/2}\left[(\cos x)^n\, _2F_1\left(-\frac{n}{2},-n-m+1;\frac{1}{2}-n;-\frac{16m c}{\cos^2x}\right)\right], $$ where $n$ is a positive even ...
3
votes
2answers
208 views

Polylogarithm - derivative with respect to order

Does anybody know where I could find the expression for $$\frac{\partial}{\partial s}\mathrm{Li}_s(z)\bigg|_{s=0}$$ or something similar?
0
votes
0answers
83 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
2
votes
1answer
45 views

Identity relating hypergeometric function and Legendre polynomial

In my notes I have written down the following relation: $_2F_1(a,a+\frac{1}{2};c;z)=2^{c-1}z^{(1-c)/2}(1-z)^{-a+(c-1)/2}L_{2a-c}^{1-c}\big(\frac{1}{\sqrt{1-z}}\big)\ ,$ where $_2F_1(a,b;c;z)$ is the ...
6
votes
1answer
77 views

Integral representation for Fibonacci's numbers

We know that, for example, the Gamma function is a perfect integral representation for the factorial $n!$ for a natural number $n$. $$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$ ...
64
votes
4answers
5k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
1
vote
1answer
23 views

proof the derivate of gamma function using the limit definition

using $\Gamma(z+1)=z\Gamma(z)$ and $\Gamma(z)=\lim\limits_{n\to+\infty}\frac{n!n^z}{z(z+1)\cdots(z+n)}$ proof that $$\psi(z+1)=-\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{1}{m}-\ln ...
2
votes
1answer
70 views

Relation between these expressions involving the Hypergeometric function and the Gegenbauer polynomials

I would like to find the relation between the solutions of a differential equation obtained by two different authors. The first solution is given in terms of the hypergeometric function $_2F_1$: ...
3
votes
4answers
46 views

Derivation of a function over $\frac{1}{\sinh(t)}\frac{d }{dt}$

I can not calculate the next derivative, someone has an idea $$\left( \frac{1}{\sinh(t)}\frac{d }{dt} \right)^n \left( e^{z t} \right)$$ Where $n\in \mathbb N$, $t>0$ and $z\in \mathbb C$. Thanks ...
4
votes
1answer
166 views

How to solve the Brioschi quintic in terms of elliptic functions?

Given the Brioschi quintic $$w^{5}-10cw^{3}+45c^{2}w-c^2=0$$ I'm interested in seeing different ways of solving it in terms of elliptic functions or theta functions.
1
vote
0answers
69 views

The integral of the product of three Meijer-G-functions

Is there any expression for the integral of the product of three Meijer-G-functions, where the domain of integration is $[0,1]$?