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

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27
votes
0answers
304 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
22
votes
0answers
239 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 ...
16
votes
0answers
192 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
14
votes
0answers
398 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
12
votes
0answers
192 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 ...
12
votes
0answers
474 views

Integral involving Complete Elliptic Integral of the First Kind K(k)

I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus: ...
11
votes
0answers
123 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
11
votes
0answers
156 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 = ...
10
votes
0answers
273 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
9
votes
0answers
238 views

Is this similarity just a coincidence?

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 $\tan x$. But if we do the same only in one direction, we get ...
9
votes
0answers
110 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 ...
9
votes
0answers
420 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
9
votes
0answers
213 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
9
votes
0answers
703 views

How to compute this integral of Bessel functions?

I have $\alpha_\max$ a real number between $0$ and $\frac\pi2$. Furthermore $\zeta$ and $\xi$ are positive real numbers. Now I would like compute the integral $$\int_0^{\alpha_\max} \mathrm{e}^{i ...
9
votes
0answers
328 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda ...
8
votes
0answers
87 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
8
votes
0answers
165 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
8
votes
0answers
338 views

Contour integral representation of Confluent Hypergeometric Function

My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$: From ...
8
votes
0answers
328 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
7
votes
0answers
129 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
7
votes
0answers
179 views

Second derivative of Hypergeometric function

I'm looking for the following second derivative $$ \kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = ...
7
votes
0answers
150 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...
6
votes
0answers
45 views

How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
6
votes
0answers
68 views

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions. However, it does not explain why this is the case. An elementary function is "a ...
6
votes
0answers
57 views

Function preserving exponentiation

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
6
votes
0answers
125 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
6
votes
0answers
94 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
6
votes
0answers
246 views

Identity involving $\zeta(3)$

This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
6
votes
0answers
208 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
6
votes
0answers
131 views

relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
5
votes
0answers
40 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
5
votes
0answers
144 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
5
votes
0answers
101 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
5
votes
0answers
68 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
5
votes
0answers
133 views

Integral of a product of five Bessel functions of order $0$

Does the following integral have a closed form? $$ \mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx. $$ I know that some similar ...
5
votes
0answers
152 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
5
votes
0answers
299 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
5
votes
0answers
425 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...
5
votes
0answers
139 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
5
votes
0answers
153 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
5
votes
0answers
146 views

Tricky integral $\int_{a}^{b}\frac{\gamma d \gamma}{\gamma + \phi_{1}(\mu)-e^{-\frac{\phi_{2}(\mu)}{\gamma}}}$

in this integral $a=\psi_{1}(\mu), \ b=\psi_{2} (\mu)$. I expanded the function in Taylor series (3 terms) around ($\gamma= \frac{b}{2}$), numerically (for varioud values of $\mu$, and other constants ...
5
votes
0answers
165 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
4
votes
0answers
35 views

Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
4
votes
0answers
71 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
4
votes
0answers
34 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
4
votes
0answers
106 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
4
votes
0answers
73 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
4
votes
0answers
50 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*} ...
4
votes
0answers
27 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
4
votes
0answers
59 views

Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$ S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots $$ have a closed form in terms of known constants? Neither the inverse ...