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
0answers
71 views

How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
9
votes
2answers
278 views

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
8
votes
1answer
154 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
6
votes
1answer
211 views

Prove ${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits. ...
1
vote
1answer
67 views

Meaning of function $f(x) = [x]$

What does the function $f (x) = [x]$ mean? How is it different from least and greatest integer function ?
0
votes
2answers
107 views

Extremely tough indefinite integral

This integral does indeed use special functions, so do include them here. Evaluate: $\int \frac{1}{\sqrt{x}\ln(x)} dx$ $x = {\sqrt{x}}^{2} \space \text{let} \space u = \sqrt{x}$ $= 2\int ...
0
votes
1answer
33 views

Derivatives and Integrals of Summations

Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite ...
20
votes
2answers
513 views
21
votes
2answers
418 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
3
votes
0answers
41 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
7
votes
1answer
118 views

Evaluate $\int_{0}^{1} \frac{\left[\rm{Li}_2\left(\frac{1}{2} \right)-\rm{Li}_2\left(\frac{1 + x}{2}\right)\right]\ln( 1 - x)}{1 + x}\,dx$

$\def\Li{{\rm{Li}}}$How to evaluate the following integral$${\large\int_0^1} {\frac{{\left[ {\Li_2\left( {\frac{1}{2}} \right) - \Li_2\left( \frac{1 + x}{2} \right)} \right]\ln \left( {1 - x} ...
1
vote
1answer
73 views

On the convergence rate at infinity of the Fourier transform of the standard bump function

My question is concerned with the Fourier transform of the standard bump function, $\phi(x)=e^{-\frac{1}{x^2-1}}$ if $x\in (-1,1)$ and equal to $0$ if otherwise. As known, as $\phi$ has compactly ...
0
votes
1answer
28 views

dense subspace of $L^2(\Omega\times(0,T))$

I am trying to prove that the functions $f(\omega,t)=g(\omega)h(t)$ where $g\in G,\: h\in H,$ are dense in $L^2(\Omega\times(0,T))$ if $G$ is dense in $L^2(\Omega)$ and $H$ is dense in $L^2((0,T))$. ...
4
votes
2answers
82 views

How to calculate the value of the special integral

I get $${\left. \frac{\partial ^2}{\partial n^2} \left( \frac{\partial ^2}{\partial m^2} B(m,n) \right) \right|_{m = \frac{1}{2},n = 0}} = \int_0^1 \frac{\ln^2 x \ln^2 (1 - x)}{\sqrt x (1 - x)} \, dx ...
2
votes
0answers
108 views

Estimating a special exponential integral

Let $s>0$ be fixed, and consider for $p>0$, $\alpha>0$, the integral $$I_s(p,\alpha)= \int_0^1 t^p e^{-s\left(1-t^2\right)^{-\alpha}} dt $$ For fixed $\alpha$, one has $I_s(p,\alpha)\to0$ ...
0
votes
1answer
38 views

Name of special function used used by Wolfram integrator

Integrating $\frac{e^{-r}}{(\sqrt{2t-r})}$ with respect to $r$ between $r=t$ and $r=2t$ on Wolfram (http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605) gives the answer ...
0
votes
0answers
27 views

How to sketch the graph of $y=6-4\cos(x/2 )$?

How can I sketch a graph such as $y=6-4\cos(x/2 )$? Should we stretch the graph horizontally with a scale factor of $2$, translate the graph by $6$ units in the positive $y$-direction and the graph ...
0
votes
0answers
14 views

What is the closed form for generation function of $\xi(2x)$ (Riemann Xi)?

I wonder whether the following coincidence is just random. Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. ...
2
votes
2answers
72 views

An infinite series involving Legendre polynomials

For $x \in [-1,1]$ and $0 \le g < 1$, consider the convergent series $$ H(x,g) = \sum_{k = 0}^\infty (2k+1) g^k P_k(x)^2 $$ where $P_k$ is the $k$-th Legendre polynomial. Then $H(1,g) = ...
0
votes
0answers
16 views

Notation for operator that returns square of a function?

Let $F$ denote the vector space of all real-valued continuous functions on the real line. Suppose I have an operator $T:F\to F$ such that for any input function $f \in F$, $T$ returns the square ...
2
votes
0answers
28 views

Estimates of certain exponential series

I am interested in series of the form $$S(k)=\sum_{n=k}^\infty e^{-an^\alpha},$$ where $a>0$ and $0<\alpha\leq1$ are fixed parameters. Clearly, this series converges, i.e. $S(k)\to 0$ for ...
1
vote
1answer
32 views

How to determine a floor function is inverse or not?

from Z to Z: f(n)=2*floor(n/2) How to determine if this function one-to-one and onto? In other words, how to determine a floor function is inverse or not?
1
vote
0answers
26 views

Is there a finite set comprising the solutions to indefinite integrals of common functions?

There are some integrals that are impossible to express in terms of elementary function, for example, $ \int \frac{e^x}{x} dx $ is only expressible as a "special" function $Ei(x)$, the exponential ...
5
votes
3answers
1k views

Integral Representations of Hermite Polynomial?

One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
11
votes
1answer
167 views

Compute polylog of order $3$ at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order $3$ at $\frac{1}{2}$ or $\operatorname{Li}_3\left(\frac{1}{2}\right)$, but how ...
3
votes
0answers
25 views

Given $n$, find $a,b$ such that $a+b=n$ and $\Omega(a)+\Omega(b)$ is maximized

Given a number $n$, find $a,b$ such that: $a,b$ non-negative integers $a+b=n$ $\Omega(a)+\Omega(b)$ is maximized $\Omega(n)$ counts the number of prime factors of n (with multiplicity). ...
1
vote
0answers
34 views

How are Fox-H functions useful in math?

The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known ...
23
votes
2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
0
votes
1answer
24 views

Image of $f$ in $f(x)=\lfloor x\rfloor$ out of bounds for intervals?

Edit 1. This all being worked on with the real numbers $\mathbb{R}$ Given a function $f(x) = \lfloor x\rfloor$ (Floor function). Find the image of B, $f^{-1}(B)$ if $B = [0,1)$ For easier cases such ...
1
vote
0answers
36 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
0
votes
1answer
17 views

The Value of One Function Determines the Value of Another

The value of $\pi(s)$ determines the value of $m(n)$. How do we describe such a relationship between two functions in standard terminology? How do we express this mathematically?
2
votes
0answers
25 views

What is the closed form of certain sum in Mathematical Epidemiology?

The following sum appears in Mathematical Epidemiology in the context of the schistosomiasis: $$\sum _{p=0}^{\infty } \left( \sum _{q=0}^{\infty }{\frac {\min \{ p,q \} {\lambda}^{p+2+q}}{ \left( p+2 ...
2
votes
1answer
608 views

Elliptic integrals with parameter outside $0<m<1$

I'm attempting to implement an equation (for calculating magnetic forces between coils, eqs (22–24) in the linked paper) that requires the use of elliptic integrals. Unfortunately these equations ...
3
votes
1answer
78 views

What is the residue of the reciprocal of Klein's $j$-invariant $1/j(\tau)$ at $e^{2\pi i /3}$

In Mathematica, Residue[1/KleinInvariantJ[t], {t,Exp[2*pi*I/3]}] results in a very ungainly expression involving complete elliptic integrals of the first and ...
2
votes
2answers
39 views

how can I derive this identity involving the integral representation of Digamma function

In here, there is an identity in equation 17 $$ k {N-1\choose{k}}\int_0^1 p^{k-1}(1-p)^{N-k-1}\log p\,dp =\psi(k)-\psi(N), $$ where $N$, $k(<N)$ are integers and $p$ can be regarded as the ...
5
votes
0answers
80 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
1
vote
2answers
59 views

What is the closed form of a certain sum

The following sum appears in a problem of Mathematical Epidemiology: $$P(m)=\sum _{p=0}^{\infty } \left( \sum _{q=0}^{\infty }2\,{\frac {\min \{ p,q \} {{\rm e}^{-m}} \left( \frac{m}{2} \right) ...
0
votes
0answers
18 views

Correlation method for speed measurement

I have data from two sensors on the road and I'd like to calculate the vehicle speed. The data are a bit noisy, so I was thinking about making a correlation of these data from two sensors with a ...
0
votes
1answer
38 views

Generating function for a binary sequence

I don't know this subject so my question may not be expressed in the accurate form. Is there a function, or a structure, that generates any desired sequence of 0 and 1 of length n? Assume we can pad ...
0
votes
0answers
25 views

Solving a transcendental function using the Lambert function

The solution to the equation $$Xe^X=K$$ is given by $$X=W(K)$$ where $W$ is the Lambert function. This idea was extended here to show that the solution to $$\frac{1-e^X}{X}=K?$$ is given by ...
2
votes
0answers
33 views

Digamma equation identification

I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ...
16
votes
2answers
335 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
3
votes
1answer
43 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
8
votes
1answer
233 views

Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?

Background I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a ...
3
votes
2answers
134 views

Evaluate $\int_{0}^{\frac {\pi}{3}}x\log(2\sin\frac {x}{2})\,dx$

Prove $$\int_0^{\pi/3}x\log \left(2 \sin\frac {x}{2}\right)\,dx = \frac {2\zeta(3)}{3}-\frac {\pi^2}{9}\log (2\pi)+\frac {2\pi ^2}{3}\log \left|\frac {\Gamma_2 \left(\frac {5}{6}\right)}{\Gamma_2 ...
12
votes
0answers
143 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
1
vote
2answers
42 views

How to derive the hyperbola giving the foci and the fixed differene

Given the two foci coordinates $(x_1,y_1)$ and $(x_2,y_2)$ of the hyperbola and the fixed difference distance, how can I derive its function to be able to draw it.
1
vote
0answers
18 views

Convergence of a Double Sum over 2 integers

Does the following double summation over x, x' (both integers) converge? $\sum\limits_{x=-\infty}^\infty \sum\limits_{x'=-\infty}^\infty \frac{Sin^2(2 \pi(x-x'))}{(x-x')^2}$. If so evaluate the sum. ...
13
votes
2answers
512 views

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

I am asking this question out of curiosity. $$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
5
votes
1answer
55 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...