Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.

learn more… | top users | synonyms

0
votes
1answer
1k views

invert complete elliptic integral of first kind K(k)

Hi, I am not a mathematician although I use it in hydrodynamics research. I have a question regarding elliptic integrals for my research in wave theory Given the value of the complete elliptic ...
5
votes
2answers
471 views

A problem on sums of arctangents of rationals

Let $S$ be a set of rational numbers. For "special" sets $S$, we can ask if $\pi$ can be written as a $\mathbb{Q}$-linear or $\mathbb{Z}$-linear combination of elements from '$\{\tan^{-1}(x): x \in ...
7
votes
3answers
866 views

Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics. The classical Schwarz Christoffel theorem does the job ...
8
votes
1answer
770 views

Is this sequence of polynomials well-known?

While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ...
7
votes
0answers
566 views

On Stark's conjecture for imaginary quadratic fields

In the famous paper "L-Functions at s = 1. IV. First Derivatives at s = 0" of Stark from 1980, it is shown that in the case of an imaginary quadratic field $K$ certain numbers of the form ...
1
vote
2answers
418 views

Bessel functions in an Hermite-Gauss basis

┬┐Could somebody tell me how can i write a zero order bessel function in an Hermite-Gauss basis? Thanks
4
votes
2answers
405 views

Is there a generalization of Floquet theory to elliptic functions?

Hi, Consider a system of linear differential equations $$ {d f \over dz} = A(z) f, $$ where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, ...
2
votes
0answers
448 views

Generalization of repeated error function integral

Is there a name for the following integral? $f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$ The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of ...
0
votes
2answers
651 views

Modified and unmodified Bessel functions of the second kind

There's a simple relationship between $J_\nu$, Bessel functions of the first kind, and $I_\nu$, modified Bessel functions of the first kind, namely $I_\nu(z) = i^{-\nu} J_\nu(iz)$. However, there ...
4
votes
2answers
1k views

Are there any uses for complex sine? [sin z]

The sine function can take a complex argument. e.g. sin(x + iy) But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am ...
1
vote
1answer
209 views

evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral, $$ {1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi} \sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right) \sin^{2}\left(\theta_{2} \over 2\right)\,} \,{\rm ...
5
votes
2answers
656 views

Proving a hypergeometric function identity

While playing around with the fractional calculus, I got stuck trying to show that two different ways of differintegrating the cosine give the same result. DLMF and the Wolfram Functions site don't ...
14
votes
3answers
1k views

nth-order generalizations of the arithmetic-geometric mean

The arithmetic-geometric mean, $a_{k+1}=\frac{a_k+b_k}{2} \quad b_{k+1}=\sqrt{a_k b_k}$ is one of the celebrated discoveries of Gauss, who found out that it is equivalent to computing a (complete) ...
-3
votes
1answer
612 views

Gamma(e)=Pi/2,Zeta(e)=4/Pi ? [closed]

I find that Gamma(e) is close to Pi/2 and Zeta(e) is close to 4/Pi. So I have a question: $\Gamma (e) = \pi /2$ $\zeta (e) = 4/\pi $ Is it true in fact?
5
votes
3answers
866 views

Differential equation for a ratio of consecutive Bessel functions

My attempts to search via Google seem to be failing, so I thought of asking here. All the derivatives of the function $r_n(z):=\frac{J_n(z)}{J_{n-1}(z)}$ where $J_n(z)$ is the Bessel function of ...
5
votes
1answer
1k views

When is ArcTan a rational multiple of pi?

Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$? Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which ...
4
votes
3answers
2k views

Finding a recursion for a sum of Legendre polynomials

The polynomial $a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$ where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature. I am ...
4
votes
2answers
620 views

Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions. I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...
0
votes
1answer
450 views

modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder $R_{k,m} \equiv H_{k} ~ \mod H_m$ for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...
1
vote
1answer
548 views

Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral $\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ ...
2
votes
1answer
328 views

On the elliptic logarithm and elliptic exponential

Browsing the wealth of functions available in Mathematica, one encounters two not-so-common functions: the "elliptic logarithm", which is an elliptic integral in another garb, and the "elliptic ...
5
votes
3answers
943 views

Simultaneously computing a complete elliptic integral and its complement

The complete elliptic integral of the first kind $K(m)=\int_0^{\pi/2}\frac{\mathrm{d}t}{\sqrt{1-m\sin^2t}}$ is easily computed via the arithmetic-geometric mean iteration; to wit, ...
4
votes
3answers
2k views

Spherical Harmonics - a bunch of questions about them

Hi there, Please tell me if I should divide these into individual questions next time. Short intro: Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...
26
votes
0answers
1k views

Curious $q$-analogues

Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$ and the Lucas polynomials $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...
7
votes
3answers
1k views

Recent work on hypergeometric functions

Does anyone know of a monograph/survey on the modern history of (basic or elliptic) hypergeometric functions and their applications? I haven't had much time to search the literature, and because it ...
18
votes
0answers
1k views

Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation $$ [m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}. $$ Consider the following trigonometric ...
1
vote
0answers
805 views

Trigonometric identities and (several?) complex variables

I don't know anything about several complex variables nor whether that topic will answer my questions below, but in one complex variable one learns that since $\sin x$ and $\cos x$ are entire ...
3
votes
1answer
926 views

Trigonometric identities

In the rant I wrote at http://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi I asked: Are these four identities the first four terms in a sequence that continues? This ...
7
votes
1answer
1k views

Errata for Emil Artin's 'The Gamma Function'?

In the English translation of The Gamma Function by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24: $$\Gamma(x) = ...
1
vote
0answers
644 views

Bessel function in polar coordinates

I want to write the Bessel function of the first kind in polar coordinates $J_\alpha(z)=|J_\alpha(z)|e^{i\varphi_\alpha(z)}$ Is anything known about $\varphi_\alpha(z)$? In particular, I'm ...
1
vote
3answers
569 views

A trigonometry problem

Let x = pi/(2k+1), for k>0. Prove that cosxcos2xcos3x...coskx = (1/2)^k I've confirmed this numerically for n from 1 to 30. I'm finding it surprisingly difficult using standard trig formula ...
7
votes
2answers
476 views

Relation between full elliptic integrals of the first and third kind

I am working on a calculation involving the Ronkin function of a hyperplane in 3-space. I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical ...
3
votes
3answers
3k views

Finding the length of a cubic B-spline

Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are chorded approximations and the opinion that the analytic solution is "unbearably gruesome". ...
15
votes
5answers
9k views

Inverse gamma function?

This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this. We have the ...
8
votes
3answers
792 views

No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads: Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2) On ...