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

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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
65 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 ...
3
votes
0answers
65 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
0answers
63 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
51 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? ...
4
votes
2answers
121 views

Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for ...
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 ...
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 ...
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
65 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
66 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
57 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 ...
6
votes
1answer
108 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
71 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
80 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)}\, ...
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 ...
2
votes
2answers
39 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
1answer
42 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 ...
7
votes
1answer
135 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 ...
10
votes
1answer
190 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 ...
51
votes
6answers
5k views

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

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 $f(x)=e^{x^2}$, then the derivative is easily ...
2
votes
0answers
64 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.
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
28 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
85 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
61 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
50 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 ...
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 ...
2
votes
0answers
37 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 ...
1
vote
0answers
51 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 ...
0
votes
1answer
40 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: $$\Phi(x) = \frac12 +\frac{1}{\sqrt{2\pi}}\int_0^x e^{-t^2/2}\,dt$$
3
votes
0answers
24 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
26 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
3
votes
1answer
53 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 ...
1
vote
1answer
51 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 ...
6
votes
1answer
89 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 $n$. $$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$ ...
7
votes
1answer
150 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 ...
1
vote
1answer
28 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 ...
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 ...
2
votes
0answers
61 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity ...
1
vote
1answer
31 views

Integral representation of the modified Bessel function involving $\sinh(t) \sinh(\alpha t)$

I've come across this peculiar integral representation for $K_\alpha(x)$: $\frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)}$ How does it come about? Are there ...
3
votes
1answer
108 views

Special functions related to $\sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$

While doing some caculation related to von Neumann entropy, I encountered this kind of convergent series. $$\text{Exl}(x) \equiv \sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$$ In my calculation, ...
0
votes
0answers
27 views

Seeking a closed form solution to an ODE, if any.

I am try to solve the following ODE by the method of separation of variables. $$\frac{dF}{dx}=\frac{b}{x\left\{\ln^3 x+\ln x[c+d(ax+b)][2\ln x+c+d(ax+b)] \right\}},$$ where $a,b,c,d$ are constants. I ...
1
vote
0answers
37 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
0
votes
0answers
11 views

Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions ...
0
votes
0answers
40 views

Relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

What are the relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$. Specifically, I need a transformation that transforms: $_2F_1\left(a,b;c; -\sinh^2(x)\right)$ to ...