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

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87
votes
9answers
5k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is ...
71
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
54
votes
1answer
1k views

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

Consider the following integral: $$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$ where $J_\mu(x)$ is the Bessel function of the first kind: ...
47
votes
4answers
2k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
46
votes
2answers
502 views

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is ...
40
votes
1answer
1k views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
38
votes
1answer
1k views

Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

Is it possible to simplify this expression? $$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$ Is there a systematic ...
37
votes
3answers
1k views

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...
37
votes
4answers
3k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
34
votes
9answers
9k views

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ ...
33
votes
3answers
608 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ \mathrm dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ ...
32
votes
1answer
543 views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
31
votes
1answer
682 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
30
votes
1answer
586 views

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
30
votes
2answers
4k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
30
votes
2answers
1k views

Is this function decreasing on $(0,1)$?

While doing some research I got stuck trying to prove that the following function is decreasing $$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$ for $k \in (0,1)$. ...
29
votes
1answer
381 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
28
votes
3answers
545 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
28
votes
3answers
774 views

$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

Please help me to solve this integral: $$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$ I managed to calculate an indefinite integral of the left part: $$\int\frac{\cos x}{\sin ...
27
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
27
votes
6answers
529 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
26
votes
4answers
2k views

When is an elliptic integral expressible in terms of elementary functions?

After seeing this recent question asking how to calculate the following integral $$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$ and some of the comments that suggested that it was an ...
26
votes
1answer
462 views

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in ...
26
votes
1answer
501 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
26
votes
1answer
665 views

elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
25
votes
7answers
2k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
25
votes
1answer
572 views

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
25
votes
2answers
541 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
23
votes
4answers
1k views

Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 ...
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( ...
22
votes
4answers
964 views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
22
votes
2answers
429 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 ...
21
votes
3answers
401 views

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

I need some help with solving this integral involving Bessel function: $\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
21
votes
2answers
419 views

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

I need to find a closed form for these nested definite integrals: $$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$ The inner integral can be ...
21
votes
1answer
531 views

Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
21
votes
0answers
457 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
20
votes
4answers
624 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
20
votes
2answers
529 views
20
votes
1answer
350 views

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
20
votes
1answer
681 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $$ analytically. We can also write $$ ...
20
votes
1answer
429 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
20
votes
1answer
241 views

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
20
votes
1answer
512 views

Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
20
votes
1answer
574 views

Intuition why the volume and surface area of the unit sphere eventually decrease

The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$ both attain maximum values for finite $n$. We can ...
20
votes
1answer
466 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
19
votes
4answers
504 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
19
votes
3answers
417 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
19
votes
2answers
565 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= ...
19
votes
2answers
396 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
19
votes
1answer
290 views

A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

I am stuck with this integral: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx,$$ where $J_0$ is the Bessel function of the first kind. Is it possible to express this ...