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Questions tagged [special-functions]

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

15
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0answers
189 views

A “relative” of the exponential function

It is well known that $$\mathrm{e}=\sum_{n=0}^\infty \frac{1}{n!}.$$ Recently, I was reminded that the volume of an $n$-simplex in $n$-dimensional space, with vertices $v_0,v_1,\dots,v_n$ is $$\left| ...
15
votes
0answers
389 views

Is this similarity just a coincidence?

Here is a graph of the function $y=-1/x$: 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 ...
14
votes
0answers
931 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: $$\int_0^{\sqrt{\frac{\alpha}{1+\...
13
votes
0answers
532 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
12
votes
0answers
653 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} \sum_{z\in\{...
12
votes
0answers
305 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 ...
11
votes
0answers
220 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 ...
11
votes
0answers
389 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic $C^\...
11
votes
0answers
695 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 K.T....
11
votes
0answers
1k 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 \...
10
votes
0answers
300 views

How to generalize Reshetnikov's $\arg\,B\left(\frac{-5+8\,\sqrt{-11}}{27};\,\frac12,\frac13\right)=\frac\pi3$?

We have, $$\arg z_1 = \frac{k\,\pi}3, \quad z_1 = \left(\tfrac{1+\sqrt{-3}}{2}\right)^k\tag1$$ $$\arg z_2=\frac{k\,\pi}3, \quad z_2 = \left( B\Big(\color{blue}{\tfrac{-5+8\,\sqrt{-11}}{27}};\,\tfrac12,...
10
votes
0answers
220 views

Bear of an integral

I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it. $$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \...
10
votes
0answers
214 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 ...
10
votes
0answers
512 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 (...
9
votes
0answers
159 views

About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
9
votes
0answers
464 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 \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\...
8
votes
0answers
287 views

An $\operatorname{erfi}(x)e^{-x^2}$ integral

I want to find an elementary evaluation of $$I=\int_0^\infty \left(\frac{\sqrt\pi}2\operatorname{erfi}(x)e^{-x^2}-\frac1{1+2x}\right)dx$$ where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^...
8
votes
0answers
322 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
186 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\Big(\frac1{\phi^6}\Big)-4\text{Li}_2\Big(\frac1{\phi^3}\Big)-3\text{Li}_2\Big(\frac1{\phi^2}\Big)+6\text{Li}_2\...
8
votes
0answers
592 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
8
votes
0answers
143 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 $\operatorname{SL}(2,\mathbb{...
7
votes
0answers
118 views

Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
7
votes
0answers
107 views

Solution to Vector Lambert W function type Equation

I was wondering if anyone has any ideas for a closed-form solution to the equation $$Ax + \exp(x) +b =0$$ where $x,b \in \mathbb{R}^n$, $A$ is a symmetric positive definite matrix and $\exp$ denotes ...
7
votes
0answers
145 views

Dilogarithmic fashion: the case $(p,q)=(3,4)$ of $\int_{0}^{1}\frac{\text{Li}_p(x)\,\text{Li}_q(x)}{x^2}\,dx$

Today I met on Facebook the following integral $$\int_{0}^{1}\frac{\text{Li}_3(x)\,\text{Li}_4(x)}{x^2}\,dx$$ which I proved to be equal to $$\frac{10 \pi ^2}{3}-\frac{17 \pi ^4}{180}-\frac{\pi ^6}{...
7
votes
0answers
192 views

Generating function of the sequence $\binom{2n}{n}^3H_n$

Generating functions of the sequences $\binom{2n}{n}^2H_n$ and $\binom{2n}{n}^2H_{2n}$, where $H_n$ is $n$-th harmonic number, are known in terms of elliptic integrals $$ \sum_{n=1}^\infty\binom{2n}{n}...
7
votes
0answers
321 views

Is there a closed form for $\,_4 F_3(1,1,1,3; 3/2,5/2,5/2;1)$?

A semi-algebraic generalization of the Steiner surface has appeared, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...
7
votes
0answers
139 views

Imaginary part of expression involving complete elliptic integral

Suppose an expression $$ \tag 1 G(z) = \frac{1}{\sqrt{(z-1)^{3}(z+3)}}\text{K}\left( \sqrt{\frac{16z}{(z+3)(z-1)^{3}}}\right), $$ where $$ K(x) \equiv \int \limits_{0}^{\frac{\pi}{2}}\frac{dy}{\sqrt{1 ...
7
votes
0answers
221 views

definite integral of elliptic integral of first kind

The signal-to-noise ratio of a Hall-effect magnetic sensor is proportional to $$ H(f,p)=\frac{I_1 (f,p)}{\sqrt{KK'(\frac{1-f}{1+f})} \sqrt{KK'(\frac{1-p}{1+p})}} $$ with $KK'(x)=K(x)K'(x)$ and $K'(x)=...
7
votes
0answers
109 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 ...
7
votes
0answers
121 views

Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
7
votes
0answers
275 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
154 views

Evaluation of $\int_0^1\frac{\ln(x)\ln(x+1)\ln(x^2+x+1)}{(1-x)(1+x^2)}dx$

I originally saw this logarithmic integral pop up on the Integrals and Series forum but due to the inactivity there no conversation has developed. I originally tried generalizing the integral to $$I(...
6
votes
0answers
326 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
6
votes
0answers
161 views

How to compute the following integral?

Someone has an idea to calculate the following integral, for $a,b,\alpha \in \mathbb{R}$, $$I_{a,b,\alpha} = \int_{1}^{+\infty} e^{-at} \,\left(1-t^{-1}\right)^b \log^{\alpha}\left(1-t^{-1}\right)...
6
votes
0answers
213 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
6
votes
0answers
821 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
6
votes
0answers
180 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 ...
6
votes
0answers
114 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*} B(\lambda)&=&\frac{1}{2\pi}\int_{0}^{2\pi}\sqrt{\lambda^2+1-2\lambda\cos\theta}\,d\theta=\...
6
votes
0answers
229 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 ...
6
votes
0answers
200 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( ...
6
votes
0answers
499 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} \...
6
votes
0answers
232 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 ...
6
votes
0answers
206 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)} + \frac{...
5
votes
0answers
73 views

Is there an identity between the Clausen function $\rm{Cl}_8\left(\frac\pi3\right)$ and $\sum_{n=1}^\infty \frac{1}{n^9\,\binom {2n}n}$?

Given the log sine integral, $$\rm{Ls}_m\Big(\frac{\pi}3\Big) = \int_0^{\pi/3}\Big(\ln\big(2\sin\tfrac{\theta}{2}\big)\Big)^{m-1}\,d\theta$$ we have in this post, $$\begin{aligned} \frac\pi{2!}\,\...
5
votes
0answers
159 views

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
5
votes
0answers
104 views

Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?

Can $\,\displaystyle\int\frac{\text{Li}_2(x)^2}{x}dx\,$ be calculated by a sum/term of polylogarithm functions and the natural logarithm and polynomials (“closed form”) ? For the special case $\,\...
5
votes
0answers
65 views

What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
5
votes
0answers
59 views

Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$

I'm trying to obtain a closed form expression for the following definite integral in $\mathbb{R}^2$: $$ \int_{0}^{\pi} \int_{u = 0}^{u= A \sin\theta} f(\theta, u)\,\mathrm{d}u \,\mathrm{d}\theta ~,\...
5
votes
0answers
162 views

Theory of ordinary differential equations - challenging problem (partially solved) - request for help

Preliminary notation: $\wedge$ - logical "and", $J$ - bessel function of I kind, $Y$-bessel function of II kind Problem: a) Solve: $\frac{d^{2}y}{dx^{2}} - [\frac{p(p+1)}{x^{2}} + c]y = 0$ $(\star)$ $...
5
votes
0answers
81 views

The choice of contour in the definition of Meijer G

It appears that when the Meijer G function is discontinuous on the unit circle, the integrals over the left and the right loops can exist but differ. For $G_{1,1}^{0,1}\left(z\,\middle|\begin{array}{c}...