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

learn more… | top users | synonyms

1
vote
1answer
53 views

Can this series be expressed as a Hyper Geometric function

I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma ...
0
votes
1answer
35 views

How to explain hypergeometric $2F_1[1+m,n,2+m,-2]$?

Question as title showed. What expression it represents? Many thanks.
3
votes
1answer
99 views

Simplification of Kampé de Fériet function

I was dealing with a convolution type integral $$ \int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t $$ By applying one of the identities in Exton's book, the solution ...
0
votes
0answers
17 views

Separable sigmoid-like function

Is there any squashing function that is multiplicatively separable? I'm looking for a sigmoid-like function f(x) that allows me to calculate f(ax)+f(ay) given "a" and f(x)+f(y).
0
votes
1answer
22 views

Plot my own function 'in two variables'

I have matlab function in two variables say $function_{something}(t)$ where $t$ is of size $2 \times k$, this is to allow to evaluate $k$ times in two values. So my output is then a $k \times 1$ ...
0
votes
0answers
14 views

Finding an analytic form of a function that satisfies asymptotic conditions

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so ...
0
votes
3answers
94 views

Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$

I recently ran into this series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ Of course this is just a special case of the Beta Dirichlet Function , for $s=3$. I had given the following ...
1
vote
0answers
51 views

Relation between Nuttal Q-function and Gaussian Q-function

I am trying to express the famous Nuttal Q-function, given as: $$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$ where $m$, $n$, $p$, and $q$ are constants and ...
5
votes
0answers
94 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
1
vote
1answer
382 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
2
votes
0answers
45 views

q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
0
votes
1answer
11 views

generate large range function from smaller

i lies between 0 to 31, with equal probability (1/32). k is either 0 or 1.(equal probability 1/2) How can one generate i using only k?
9
votes
3answers
675 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
9
votes
1answer
207 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
1
vote
0answers
50 views

Solving Integral $\int_{0}^{x}K_{0}\left(\frac{2t^{\beta /2}}{\sigma ^{\beta }}\right)\;dt$

This question is a continuation of the question posted here. The problem here is to solve the integral with modified Bessel function of second kind, $K_0\left(u\right)$: $$F\left ( x \right ...
4
votes
3answers
577 views

Integrating Modified Bessel function of the second kind?

I need to compute the following integral: $$\int_0^\infty\;\;K_0\left(\sqrt{a(k^2+b)}\right)dk$$ where $a>0$ and $b>0$. I have tried several substitutions and played around a lot in ...
3
votes
1answer
56 views

Identities for hypergeometric functions ${}_2F_1$ with z=1/2

Is there a closed form (or approximation) for a hypergeometric function of form: $_2F_1(1,b+c;c;\frac{1}{2}) \quad \text{where} \; b,c \in \mathbb{N}$ ? I researched all identities in ...
2
votes
1answer
33 views

Proving that $\lim \limits_{b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$

I am trying to prove that $\lim \limits_{a,b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$ . Here $F$ is the hypergeometric function. Here because of two limits I am unable to ...
9
votes
2answers
219 views

How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$

let $$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$ How find the $a_{2^n}=?$ my idea:let ...
3
votes
3answers
105 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
0
votes
0answers
18 views

Polylogarithm and unclear statement

I am trying to solve this question which may not have an answer at all, but any clarification would be much appreciated. I also tried to explain what I have tried/thought about it below. Let ...
4
votes
0answers
50 views

About a sequence related with the complete elliptic integral of the second kind

When answering this related question I proved that if we define $B(\lambda)$ as: $$\begin{eqnarray*} ...
0
votes
0answers
38 views

Proving That Two Paths of Different Lengths Are Adjoined

In the section on 'Adjoining Paths' of its 'Topology' book's page on 'Path Connectedness,' WikiBooks shows that, for any topological space $X$ with members $a$, $b$, and $c$, the following…: ...
2
votes
2answers
46 views

Dirichlet $L$ functions at $s=2$

Let $\chi$ be a Dirichlet character and let $L(\chi,s)$ denotes its Dirichlet $L$-function. What is the value of $L(2,\chi)$ ? Or simply, is $L(2,\chi)/\pi^2$ rational ? Many thanks for your answer ...
0
votes
1answer
35 views

Proving the transform of the Q-function

I have the Gaussian Q-function, given by: and I want to prove that it can be also expressed as: Can somebody help explaining how to obtain the second integral from the first?
10
votes
2answers
461 views

Prove that $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's identity.

I'm trying to derive the functional equation $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's formula: ...
7
votes
6answers
6k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
0
votes
0answers
32 views

Definite integral with Bessel functions

Show that for $Re(\lambda)>0,Re(\mu)>0$ it holds the following identity $\int_0^a x J_\lambda(2a)I_\lambda(2x) J_\mu(2 \sqrt{a^2-x^2}) I_\mu(2 \sqrt{a^2-x^2}) dx = \frac{a^{2 \lambda + 2 \mu + ...
0
votes
1answer
42 views

Does this special function exist?

Is it possible to find a non-trivial function $f(x_1,x_2)$ that has two parameters $x_1$ and $x_2$. This function should satisfy $f(x_1,x_2) = f(\frac{x_1}{1+r},x_2 +r)$, for any non-negative $r$. ...
4
votes
1answer
389 views

ODE with additional term

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ I don't ...
0
votes
0answers
34 views

Finding general orthogonal polynomials

Many Special functions are orthogonal; for example, the sine and cosine function is an orthogonal function. Also, a couple of orthogonal polynomials are well-known. Now I'm asking the following: Given ...
2
votes
2answers
72 views

Erf squared approximation

I found a nice and useful approximation of squared error function $$ \mathrm{erf}^{2}\!\left(x\right)=1-\exp\!\left(-\frac{\pi^{2}}{8}x^{2}\right)+\varepsilon\!\left(x\right). $$ I checked ...
2
votes
1answer
21 views

Proving that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$

I am trying to prove that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$ without using dominated convergence theorem. Here $F$ is the hypergeometric function. I have been able to ...
1
vote
1answer
31 views

inequality based on Hermite polynomial

How to prove that $$|H_n(x)| \leq |H_n(ix)|?$$ I have tried with the explicit representation of the Hermite polynomial, but can't reach the target. Any clue please.
2
votes
3answers
563 views

What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
3
votes
1answer
32 views

Real positive zeros of the polylogarithm function

The polylogarithm function $Li_{s}(z)$ is defined as: \begin{equation} Li_{s}(z)=\sum_{k=1}^{\infty} \dfrac{z^{k}}{k^{s}}. \end{equation} My question is: do there exist any real positive zeros of ...
22
votes
1answer
459 views

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
0
votes
0answers
34 views

Is there a standard name for the functions $f(x) = |x|^q$?

Do the following sets of functions have common/standard names? $f(x) = |x|^n$ for $n \in \mathbb{N}$ (or $\mathbb{N}\cup \{0\}$) $f(x) = |x|^n$ for $n \in \mathbb{Z}$ $f(x) = |x|^r$ for $r \in ...
1
vote
0answers
78 views

Integral: product of Bessel functions

I work on a project and I am blocked on the question to determine the auxiliary density $\Omega_{\mu}(r)$ such that \begin{equation} ...
1
vote
1answer
62 views

Sum $\sum _{k=0}^{\infty } \frac{(-1)^k \psi (k+1)}{\left(k-\frac{3}{2}\right)^2}$

Is it possible to get a closed form for: $$\sum _{k=0}^{\infty } \frac{(-1)^k \psi (k+1)}{\left(k-\frac{3}{2}\right)^2}$$ where $\psi$ is the polygamma function?
1
vote
0answers
34 views

Fundamental questions about Logarithm and finding quadratic roots

Define: $(e^{iz}+e^{-iz})/2= cos z$ where $z \in \Bbb C $, i.e, the cosine function is defined for complex $z$. Now, is it true that for each $w \in \Bbb C $ there is $z \in \Bbb C $ such that $cos z ...
6
votes
1answer
117 views

About the series $\sum_{n\geq 0}\frac{1}{(2n+1)^2+k}$ and the digamma function

Let we provide a closed form for $$ S_k = \sum_{n\geq 0}\frac{1}{(2n+1)^2+k} $$ for $k>0$ in terms of elementary functions. It is quite easy to check that $S_k$ can be computed in terms of the ...
0
votes
0answers
25 views

Usage of integration representation

There are various "Integral Representations" for various functions, such as Bessel Functions. http://dlmf.nist.gov/10.9 How is this integral regpresentation used? In numerical calculation? or In ...
2
votes
1answer
41 views

H-function for the following integral

I stumbled upon the integral $\int\limits_0^{+\infty} u^\nu\exp(-au-bu^\rho)du$, $\Re(a)>0,\,\,\Re(b)>0,\,\,\rho>0$. I cannot find any way to represent it using the Fox-H function. Any hints? ...
0
votes
0answers
22 views

Complex Function With No Singularities

Suppose it is given that a function f is meromorphic (no singularities except poles) and now if in any region it is given that f has no poles also, then can I assume that f is analytic/holomorphic in ...
3
votes
2answers
937 views

Does sinc function have any special inverse function defined?

We know that $y=xe^x$ cannot be solved for $x$ using elementary functions. The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above ...
1
vote
1answer
67 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...
2
votes
1answer
30 views

Possible existence of weight function $\rho (t)$

Consider $L^2[-\pi,\pi]$. We define an inner product on this space by $$\langle f,g\rangle=\int_{-\pi}^{\pi} f(t)\overline {g(t)} \, dt \quad\to(1)$$ Suppose if we introduce a weight function ...
4
votes
1answer
117 views

Proving integral of zeroth-order Bessel function multiplied by cosine with complicated arguments

How could it be proved that $$ \int_0^\infty J_0\left(\alpha\sqrt{x^2+z^2}\right)\ \cos{\beta x}\ \mathrm{d}x = \frac{\cos\left(z\sqrt{\alpha^2-\beta^2}\right)}{\sqrt{\alpha^2-\beta^2}} $$ for $0 < ...
5
votes
2answers
152 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...