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

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3
votes
3answers
101 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 $\Gamma^{2}$ ...
2
votes
2answers
42 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 ...
3
votes
0answers
46 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
1answer
32 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?
1
vote
0answers
45 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 ...
0
votes
0answers
29 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
40 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 $r$. ...
0
votes
0answers
29 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
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 ...
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
2answers
64 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 ...
0
votes
0answers
37 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…: ...
3
votes
1answer
28 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
33 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.
1
vote
1answer
12 views

Integer functions $f_k(n)$ which return {0,1} depending on whether or not $k|n$

In computer programs and physics problems it is often nice to have mathematical functions that work when you want them to but sort of zero out when they don't apply. I'm thinking of how useful delta ...
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
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?
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 ...
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 ...
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

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 ...
2
votes
1answer
36 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? ...
1
vote
1answer
53 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 ...
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 ...
19
votes
0answers
170 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 ...
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 ...
1
vote
0answers
71 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} ...
4
votes
1answer
77 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 < ...
0
votes
0answers
13 views

Can the regularized beta function be calculated for integer values of $a$ using identities?

I've been trying to create a cumulative Student's T Distribution calculator using javascript as fun side project. I've successfully created a gamma function approximator, and from there, a beta ...
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 ...
0
votes
0answers
16 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 ?
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 $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n+z}=-\psi (z)$ ...
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 ...
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 $$\frac{dy}{dx}=\frac{1}{x^2+y^2}\\ (x^2+y^2)\frac{dy}{dx}=1\\ (x^2+y^2)dy=dx$$ ...
0
votes
0answers
12 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 ...
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 ...
0
votes
1answer
30 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
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
46 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 ...
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
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 ...
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
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 ...
3
votes
1answer
48 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 ...
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 ...
0
votes
0answers
13 views

In what ways can I extend the error function to accept complex arguments?

What are the different approaches to extending the error function to accept complex arguments? When should I favor using one approach over another?
0
votes
1answer
16 views

How to define formula for decimal places

I need to define a formula for a half unit of the smallest decimal place in unit price ($UP$), or understand if this can be defined with a formula? What I have is this $$ T_{min,max} = Amt +/- (Qty ...
0
votes
0answers
108 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) ...