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

learn more… | top users | synonyms

2
votes
1answer
20 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
29 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
544 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
29 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 ...
0
votes
0answers
34 views

Closed form expression for 3F2 with positive unit argument

Is there any closed form expression for the Hypergeometric function ${}_3F_2(-n,-n,c;-d/2-n,-d/2-n;1)$ for $n>0$ and $d>0$. The parameter $c$ can be both positive and negative.
22
votes
1answer
448 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
72 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
55 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?
19
votes
0answers
172 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
1
vote
0answers
30 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 ...
4
votes
1answer
92 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
20 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
38 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
818 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
55 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 ...
0
votes
0answers
30 views

Euler type superdivergent

Could you explain where it come from $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}| \begin{array}{c} 0 \\ 0,0,0 \\ \end{array} \right)+2 \left(\log ...
4
votes
1answer
79 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
151 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 ...
4
votes
3answers
142 views

Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$?

I have the expression $$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$ where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$. I have a feeling this should ...
7
votes
1answer
119 views

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

Trying to prove $$\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$$ I found by numerical calculation that (when $k$ goes to infinity) $$\sum_{n=1}^{k}\zeta ...
1
vote
4answers
50 views

construction of a curve connecting two points

Let $a,b,c$ be positive reals numbers. Assume $a<b$. I'm trying to construct a $C^1$ function (meaning a function with continuous derivative) $f$ with the following properties: $f$ is increasing ...
3
votes
2answers
130 views

Solve $-B \ln y -A y \ln y + A y- A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $-B \ln y -A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
0
votes
0answers
17 views

Alternative Function Definitions for the Square Wave signal

Are there any other function definitions for the Square Wave signal rather than the : and those referred to Wikipedia ?
7
votes
0answers
89 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
1
vote
1answer
24 views

An identity on Polygamma

I would like to know how to prove that: $$\psi^{(n)}(z)=(-1)^{n+1}n! \sum_{k=0}^{\infty}\frac{1}{\left ( k+z \right )^{n+1}}$$ I know that $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n+z}=-\psi (z)$ ...
0
votes
1answer
36 views

Legendre's Chi- Function

I want to get the numerical value(twenty at thirty decimals) of $$\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ Thanks you very much.
2
votes
2answers
39 views

how to compute this limits given these conditions.

if $f(1)=1$ and $f'(x)=\frac{1}{x^2+[f(x)]^2}$ then compute $\lim\limits_{x\to+\infty}f(x)$ i tried to write it was $$\frac{dy}{dx}=\frac{1}{x^2+y^2}\\ (x^2+y^2)\frac{dy}{dx}=1\\ (x^2+y^2)dy=dx$$ ...
1
vote
0answers
327 views

Fourier-Bessel series coefficients

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for $k_1$and $k_2$ both zeroes of $J_n(t)$, the orthogonality relation given by: $$\int_0^1 ...
0
votes
0answers
110 views

Integral of the product of two Normal distribution CDF (erf)

How do I solve the following? $$ \lim_{x \rightarrow \infty} \int_0^{x} \left[ 1 + \text{erf} \left( \frac{\epsilon - a}{b} \right) \right] \left[ 1 + \text{erf} \left( \frac{\epsilon - c}{d} \right) ...
0
votes
0answers
13 views

Analyticity of Mellin Barnes integral

How to decide the analyticity of Mellin-Barnes integral? In particular, When Fox's H-function is analytic? Is the condition for existence, analytic and condition for convergence both have the same ...
0
votes
0answers
10 views

Can any one tell m ewith one example, how to evaluate a double Mellin Barnes integral?

What is meant by asymptotic expansion of Gamma function? i.e. $ |\Gamma(z)| = |\Gamma(x+iy)| \approx \sqrt{2 \pi} \left|y\right|^{\left(x - \frac{1}{2}\right)} e^{-\pi \frac{|y|}{2}}, \quad ...
0
votes
1answer
30 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
3
votes
1answer
52 views

Gaussian-like integral??

It has been a long time since I've needed to do integration... hope you can help What is the result of the following where $\alpha$ is a constant; $$\int_0^\infty ...
4
votes
1answer
62 views

Series (Dilogarithm Function)

Let $\displaystyle f(x)=\sum_{n=1}^{\infty} \dfrac{x^n}{n^2} , \; x \in (0, 1)$. Evaluate $f(1/2)$ without using the known formulae of the dilogarithm or the equation it satisfies. May I have some ...
0
votes
1answer
47 views

An integral involving in Bessel, exponential and power functions

I need to solve an integral similar to the one in a book, Abramowitz and Stegun. Handbook of Mathematical Functions,P486, Eq. 11.4.29. However, I can't use infinity as the upper bound. Can someone ...
1
vote
1answer
73 views

How to bound the uniform convergence on $[0,1]$ of the Bernstein polynomials of $ e^x $ to $e^x$

I have a question: How can we prove that the Bernstein polynomial $$p_{n}(x)=\sum_{l=0}^{n} e^{l\over n}\begin{pmatrix} n\\ l \end{pmatrix}x^l(1-x)^{n-l}$$ uniformly converges $e^x$ in the interval ...
2
votes
1answer
325 views

Integral involving bessel function/gaussian/rational function

I'd like to solve: $$\int_0^{\infty}\quad J_1(ak)\,\frac{b+k^2}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}\,\exp(-ck^2)\,\,dk$$ Is there any specific rule for it? Thanks!
0
votes
1answer
20 views

Approximation of a series containing Bessel functions

I have this series: $$\displaystyle S=\sum_{k=0}^N\left(J_k(x)-J_k(y)\right)$$ where: $J_k(\dot{})$ is the Bessel function of order $k$ with $x\in\mathbb{R}$ and $y\in\mathbb{R}$. I have to calculate ...
0
votes
1answer
22 views

Evaluation of definite integral in terms of Bessel function

Can I express the integral $\int_0^1[\cos (xt)/(1-t^2)]dt$ in terms of Bessel Polynomial? I tried by putting $t=\sin \theta$ and used the integral representation of Bessel's polynomial ...
0
votes
1answer
1k views

Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
0
votes
0answers
14 views
1
vote
0answers
37 views

Can my wrong derivation of the Gamma function be fixed?

I found the following simple but wrong derivation of the Gamma function: We start from the definition of the exponential function $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} \\ \Rightarrow 1 = ...
0
votes
1answer
59 views

Definite integral involving modified bessel function of the first kind and its logarithm

I'm trying to solve the following integral $$ T=\int_0^\infty \exp(-a x^2) I_1(b x) \log(I_1(b x))\, dx $$ where $I_1(x)$ is the modified Bessel function of the first kind and order one, and $a$, $b$ ...
14
votes
3answers
322 views

Calculate the following integral $\int_0^{\pi/2} \frac{\sin^m x\,\mathrm{d}x}{\sin x + \cos x}$, $m=2k-1$

At the moment I am studing the following integral $$K(m,n)= \int_0^{\pi/2} \frac{\sin^m x\,\mathrm{d}x}{\sin^nx + \cos^nx}.$$ For integers $m$,$n$. The question regarding both $K(1,1)$ and ...
0
votes
0answers
42 views

A discrete fourier-bessel series?

A function $f$ on an interval $[0,b]$ can be expanded as a sum of Bessel functions, using the inner product $$\int_0^b f(x) g(x) x\mathrm dx$$ under which these functions are orthogonal, for example ...
0
votes
2answers
38 views

Why is the imaginary part of the logarithm of the gamma function a square wave?

I just stumpled upon it and it made me curious. Why is the imaginary part of $\ln(\Gamma(x))$ a square wave for $x < 0$ ? The square wave has a period of 2 and a amplitude of $\pi/2$. How can one ...
3
votes
1answer
50 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...