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

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1
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0answers
41 views

Manipulating constants inside the Exponential Integral function

In the following form of the Exponential Integral function; $$ E_{n}(x+c) $$ where $E_{n}(x+c)$ is the exponential integral function, $x\in\mathbb{R}^+$ , $n\in\mathbb{N}$ , $c$ is a constant Is it ...
0
votes
1answer
44 views

Asymptotic Expansions of Exponential Integral function

In NIST equation 8.20.2 what is meant by $(p)_{k}$ $$\mathop{E_{p}}\nolimits\!\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\left(p\right)_{k}}{z^{k}},$$
2
votes
1answer
45 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
4
votes
1answer
74 views

How to prove $H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2}$

How to prove: $$ H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2} $$ Where $C$ is Catalan's number, $A$ is Glaisher-Kinkelin's constant and $H(x)$ is the ...
3
votes
1answer
133 views

Closed-form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
8
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1answer
126 views

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
8
votes
0answers
60 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
4
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0answers
53 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
12
votes
3answers
265 views

Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
6
votes
3answers
84 views

Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

While I've studied integrals involving polylogarithm functions I've observed that $$\int_0^1 \operatorname{Li}_p(x) \, dx \stackrel{?}{=} \sum_{k=2}^p(-1)^{p+k}\zeta(k)+(-1)^{p+1},\tag{1}$$ for any ...
5
votes
2answers
126 views

Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ ...
3
votes
1answer
61 views

Closed-form of a special value dilogarithm identity

Let $c$ be the following. $$c = \frac{1+i\sqrt 3}{3}\operatorname{Li}_2\left(1-\frac{i\sqrt 3}{3}\right)+\operatorname{Li}_2\left(\frac 34 + \frac{i\sqrt 3}{4}\right) + ...
2
votes
0answers
19 views

Do the incomplete gamma functions have reflection formulas?

Euler gave this reflection formula for the gamma function: $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$ My question - do the lower incomplete gamma function $\gamma(s,x)$ and the upper ...
3
votes
2answers
91 views

Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$

Is there a closed-form of the following sequence? $$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$ where $_2F_1$ is the hypergeometric function and $n ...
1
vote
0answers
37 views

Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function

The reciprocal gamma function has the following Taylor series. $$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$ where the $a_k$ coefficient are given by the followint recursion. $a_1=1$, ...
10
votes
2answers
192 views

Closed-form of $\int_0^1 B_n(x)\psi(x+1)\,dx$

Is there a closed-form of the following integral? $$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$ where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function. The motivation of the ...
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) = ...
1
vote
1answer
36 views

octagonal number theorem $q$-Pochhammer symbol expression

Setting the exponents of this analogue of the series in Euler's Pentagonal Number theorem to be the octagonal numbers: $$U(q)= \sum_{n\in\mathbb{Z}} (-1)^{n}q^{n(6n-4)/2}$$ in mpmath: ...
0
votes
2answers
106 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
32 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 ...
3
votes
0answers
40 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 ...
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 ...
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. ...
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
0answers
50 views

Finding convolution identites

Suppose I have the following definition: $$\frac{x^2/2!}{e^x-1-x}=\sum_{k=0}^{\infty}A_k\frac{x^k}{k!}$$ I want to find a convolution identity for these coefficients $A_k$, but I've never studied ...
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
26 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 $\tan x$. ...
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
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) = ...
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?
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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 ...
3
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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). ...
6
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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
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0answers
33 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 ...
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 ...
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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?
2
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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 ...
0
votes
1answer
16 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?
16
votes
1answer
312 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). The problem There are $n$ different ...
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 $\operatorname{Ei}(-|x|)$. ...
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 ...
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 ...