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.

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Differential equation with some constraints

I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...
0
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0answers
249 views

High dimensional beta integral (question following the previous post)

Hello, This post is a question following the previous post. In one dimensional case, we have $$ \int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...
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2answers
442 views

High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} ...
11
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1answer
2k views

Geometric meaning of a trigonometric identity

It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then $$ a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta ...
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3answers
451 views

Pairing function monotonic respect to product of arguments [closed]

Has anyone ever created a "pairing function" (possibly non-injective) with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are ...
1
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1answer
1k views

looking for monotonically increasing functions with range in [0,1] [closed]

Ideally, the function f(x) would tend toward zero as x tends towards negative infinity, and f(x) would tend towards 1 as x tends towards infinity, all the while being monotonically increasing. for ...
7
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2answers
360 views

Asymptotics of the $q$-harmonic series as $q\to1$

The following (very simply looking!) problem occurs in regularization of the harmonic series which can be formally thought of as the limit as $q\to1$, $|q|<1$, of $$ ...
0
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0answers
217 views

Non-standard addition theorem for Legendre function of the first kind

There is known link text addition theorem for the Legendre functions of the first kind $P_{\nu}^m(x)$, which establishes the connection between $P_{\nu}^m(x)P_{\nu}^{m}(y)$ and $P_{\nu}(F(x,y))$ for ...
3
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0answers
456 views

Mathematica package for obtaining hypergeometric function

In my current research in electromagnetics I am encountering integrals of the form $$ \int_0^\infty dt J_0( r t) \frac{\exp(-h \sqrt{t^2 - a^2})}{\sqrt{t^2 - b^2}} t . $$ $a$ and $b$ are complex ...
2
votes
1answer
261 views

antiderivative involving modified bessel function

This little integral has been holding me up for weeks. Has anyone come across a similar integral in their work. $\int {\frac{d x}{c-I_0(x)}} $
1
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1answer
186 views

Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?

Dear all, I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the ...
3
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0answers
301 views

Why is Mellin-inverse of Gamma periodic?

Specific Case The periodicity is obvious from computation: $$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$ However, is there a way to see directly from the integral ...
1
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1answer
292 views

Maximum of a series of integrals of Hermite functions

Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want ...
0
votes
2answers
756 views

Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too! The original motivation for this is the ...
0
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2answers
648 views

Cosine of a Partial Sum

Does anyone know of a closed formula for $cos(\displaystyle\sum_{n=0}^m a_{n})$? I've seen formulas for $cos(\displaystyle\sum_{n=0}^\infty a_{n})$ and $tan(\displaystyle\sum_{n=0}^m a_{n})$, but the ...
4
votes
1answer
800 views

What's the difference between a Riemann theta and a Siegel theta function?

One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details. A look at the DLMF says that ...
5
votes
1answer
905 views

Geometric meaning of trigonometric relations

According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and ...
5
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1answer
683 views

Integral identity for Legendre polynomials

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial? $$ \int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1} $$ Notes & Background A variant of this ...
2
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1answer
296 views

Legendre Function “Types”

Legendre functions can be of first and second kinds; $P$, $Q$. They can have order $\mu$ and degree $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$ But do they also have Types? Some of the numerical software I am ...
7
votes
4answers
1k views

Determining the asymptotic behavior of a series

I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable ...
7
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4answers
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Estimating the probability that one Poisson RV is larger than another

Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function $$\mathbb P(X - Y ...
3
votes
3answers
2k views

Integral over error function and normal distribution

Help me understand why $\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx ...
3
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2answers
1k views

Product of hypergeometric functions/Jacobi Polynomials

Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}\_{2}F_{1}[-n,-n+1;2;x]$ and ...
22
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6answers
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Why are hypergeometric series important and do they have a geometric or heuristic motivation?

Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...
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5answers
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Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
2
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3answers
387 views

About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$. I am considering the generalized question for ...
4
votes
1answer
217 views

Positivity of a rational function

A rational function is called positive if all its Taylor coefficients are positive. Friedrichs-Lewy conjecture states the positivity of the rational function \begin{eqnarray*}\frac{1}{ (1-x)(1- ...
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0answers
269 views

monotonicity of functions related to modified Bessel function

Dear colleagues, I recently met some problems related to the modified Bessel funtions of the first kind and the second kind. I want to know if there exist some results on the monotonicity of ...
2
votes
1answer
248 views

Estimating associated Legendre function

For past months I've been trying to estimate associated Legendre function $P_{-\frac{1}{2}+it}^m (\cosh r)$ in order to study Laplacian eigenfunctions on a hyperbolic surface. I found reasonably sharp ...
0
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1answer
509 views

Results on derivatives with respect to the parameter of Modified Bessel Function [duplicate]

Possible Duplicate: Derivate Bessel Function with respect to order Dear colleagues, I have a question about the modified Bessel function of the second kind, $I_\nu(x)$ and $K_\nu(x)$. I ...
5
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2answers
534 views

generalization of (Rogers) dilogarithm

Let $C$ and $S$ be abbreviations for $\cosh$ and $\sinh$, and consider the following function: $$f(x,y) = \int_{-y\le r+l \le y} \frac{ (C(x)S(l)C(r) - C(l)S(r))(C(x)C(l)S(r)-S(l)C(r)) } ...
4
votes
2answers
372 views

Analytic continuation of $_4F_3(1)$

The Gauss theorem $${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$ allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...
6
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2answers
2k views

Sums of arctangents

$$ \begin{align} \arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\ & {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots \end{align} $$ ...
3
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1answer
375 views

Summations in $\tan^2$

Hey all, I was just wondering if anyone had come across the following identities, valid for $m\in\mathbb{N}$. I've used Abramowitz and Stegun, Maple, Mathematica etc but can't find them anywhere. I ...
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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
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2answers
476 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
923 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
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1answer
779 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
599 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
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2answers
432 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
422 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
470 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
664 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
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1answer
215 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
718 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
625 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
967 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 ...