Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

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Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express ...
Vladimir Reshetnikov's user avatar
59 votes
7 answers
4k views

How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ ...
Vladimir Reshetnikov's user avatar
29 votes
2 answers
2k views

Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

For $n\geqslant m>1$, the integral $$I_{n,m}:=\int\limits_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$$ converges. If $m$ and $n$ are both even or both odd, we can use the residue theorem to easily evaluate ...
Wolfgang's user avatar
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26 votes
2 answers
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Are these two new ways of representing odd zeta values as integrals known?

This is inspired by the same beautiful integral expression for $\zeta(3)$ as this question, but goes in a slightly different direction. Writing the original integral in the form $$\int_0^1\frac{x(1-x)}...
Wolfgang's user avatar
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25 votes
2 answers
2k views

When can an invertible function be inverted in closed form?

The Risch algorithm answers the question: "When can a function be integrated in closed form?", see: https://en.wikipedia.org/wiki/Symbolic_integration Is anyone aware of any work that answers the ...
P. Carr's user avatar
  • 351
24 votes
1 answer
2k views

Why these surprising proportionalities of integrals involving odd zeta values?

Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
Wolfgang's user avatar
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20 votes
3 answers
957 views

Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
Oksana Gimmel's user avatar
19 votes
2 answers
896 views

Could there be a special-function counterexample to Schanuel's conjecture?

It is not too hard to show that if Schanuel's conjecture is true, then the only algebraic numbers admitting a "closed-form expression" (as defined precisely in this paper) involving $e$, $\pi$, and ...
Timothy Chow's user avatar
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17 votes
1 answer
832 views

On the solvable septic quadrinomial $x^7-7x^4-14x^3-7=0$?

The concept of sparse polynomials has its place, and solvable but irreducible quadrinomial examples such as, $$x^7-7x^4-14x^3-7=0$$ $$x^8+x^7+29x^2+29=0$$ $$x^9-27x^4-9x^3-9^2=0$$ $$x^{12}-36x^5-12x^3-...
Tito Piezas III's user avatar
16 votes
3 answers
3k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
zeraoulia rafik's user avatar
14 votes
1 answer
480 views

An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series: $$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$ It can be expressed in terms of a hypergeometric function: $$S=-\frac{5}{16}\...
X.C.'s user avatar
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14 votes
1 answer
783 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. $$\gamma=-\int\...
Wolfgang's user avatar
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13 votes
2 answers
721 views

How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?

For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...
Wolfgang's user avatar
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13 votes
3 answers
793 views

Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $. Is it ...
Wolfgang's user avatar
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12 votes
2 answers
980 views

counting points on unit sphere mod p

Let $f(n)$ be the number of points on the unit sphere $x^2 + y^2 + z^2 = 1\; \pmod n$ with $x,y,z \in \mathbb{Z}/n\mathbb{Z}$ This is sequence A087784 in the Online Encyclopedia of Integer ...
john mangual's user avatar
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11 votes
5 answers
1k views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
Mark's user avatar
  • 111
11 votes
2 answers
2k views

Difficult trigonometric integral

This question was also asked here and here. I have faced some difficulties to do the following integral: $$ I=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta~\sin\theta\int_{0}^{\infty}dr~r^2\frac{3x^2y^...
Dinesh Shankar's user avatar
11 votes
1 answer
428 views

Closed Form for $\sum\limits_{n=-2a}^\infty(n+a){2a\choose-n}^4,~a\not\in\mathbb Z$

Do either $~S_4^+(a)~=~\displaystyle\sum_{n=0}^\infty(n+a){2a\choose n}^4~$ or $~S_4^-(a)~=~\displaystyle\sum_{n=-2a}^\infty(n+a){2a\choose-n}^4~$ possess a meaningful closed form expression1 in terms ...
Lucian's user avatar
  • 655
11 votes
2 answers
575 views

New results on Chow's notion of closed-form numbers?

In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen ...
Todd Trimble's user avatar
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11 votes
1 answer
609 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
Fred Hucht's user avatar
  • 2,705
10 votes
2 answers
649 views

On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

I. Some functions As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$ $$\beta(s) = \sum_{n=1}^\infty\...
Tito Piezas III's user avatar
9 votes
3 answers
522 views

Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...
joro's user avatar
  • 24.2k
9 votes
1 answer
730 views

Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1)

For positive integer $n$ the following value of a hypergeometric function $$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$ based on the first few terms looks like $$ R_1(n) + R_2(n) \pi^2$$ where $R_{1,2}(n)$ are ...
Fetchinson0234's user avatar
8 votes
1 answer
1k views

What is the value of this double sum in closed form?

I encountered the following double sum which requires an evaluation. Is there a closed form for this? $$\sum_{n=0}^{\infty}\frac{\sum_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$ Incidentally, it ...
T. Amdeberhan's user avatar
8 votes
3 answers
480 views

Invertibility of specific function

This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function: $$V\left(\varphi\right)=\frac{A\sqrt{\pi-...
Costas Vlachos's user avatar
8 votes
1 answer
987 views

Sum of subset of geometric series: a^2^n

The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is, $S = a + a^2 + a^4 + a^8 + \cdots$ ...
Henry Yuen's user avatar
  • 1,909
8 votes
2 answers
1k views

Solution to $(A+x^2)e^x=B$ with Lambert W function

Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?
DouglasKeuk's user avatar
8 votes
2 answers
577 views

On Zagier's missing continued fraction with multiple limits?

I. Zagier's continued fraction As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...
Tito Piezas III's user avatar
8 votes
0 answers
271 views

Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note: This question has been brought here from MSE. I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had ...
user avatar
7 votes
4 answers
1k views

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$ (The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv ...
Lviv Scottish Book's user avatar
7 votes
4 answers
492 views

Is this closed-form summation a special case of known Lerch zeta function formulas?

With some Poisson summation manipulations (credit: Michał Pacholski) I have convinced myself of a closed form expression for this conditionally convergent series: $$\sum_{n=-\infty}^\infty \frac{e^{in\...
Carlo Beenakker's user avatar
7 votes
3 answers
944 views

Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value: $$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$ where the $X_i \approx N(0,1)$ are iid. It seems to be an elementary ...
gappy3000's user avatar
  • 461
7 votes
1 answer
488 views

Experimental mathematics: how are floating point equations discovered/converted to exact equations?

the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
vzn's user avatar
  • 529
7 votes
1 answer
453 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
Faoler's user avatar
  • 431
7 votes
2 answers
411 views

A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics

I've always wondered if the DeMoivre method to generate an algebraic number $x_p$, $$x_p = u_1^{1/p}+u_2^{1/p}$$ of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
Tito Piezas III's user avatar
7 votes
2 answers
667 views

A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It ...
u184's user avatar
  • 277
7 votes
1 answer
406 views

Convergence and Closed Form of an Integral Involving Bell Numbers

1. Does the following integral converge ? $$\int_0^\infty \frac{b(x)}{B(x)} dx$$ where $$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$ 2. ...
Lucian's user avatar
  • 655
7 votes
1 answer
495 views

Simplifying Root of Unity Double Summation

Good afternoon. I have a particular summation, $$\zeta_{n,k}(N)=\frac{k!}{N^{n+1-k}}\sum_{j=0}^n\sum_{i=0}^{N-1}\binom{n}{j}w_N^{(j-k)i}$$ Here, the $w_N$ is the root of unity $w_N=e^\frac{2i\pi}{N}$...
Eleven-Eleven's user avatar
6 votes
3 answers
906 views

Any closed form for series like $F(x)=\sum\limits_{p=2}^{\infty}x^p,$ where $p$ is prime?

Any closed form for series like $$F(x)=\sum_{p=2}^{\infty}x^p,\quad p\text{ is prime}$$ or $$F(x)=\sum_{i=0}^{\infty}x^{i!}\quad ?$$ More generally, we can obtain a power series from decimal expansion ...
XL _At_Here_There's user avatar
6 votes
2 answers
259 views

Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
Julian's user avatar
  • 613
6 votes
3 answers
728 views

Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients $\frac{(2d)!}{(d!)^2}$ in terms of a generating function $\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$ by the Taylor series of $(1-4x)^...
user20592's user avatar
6 votes
1 answer
251 views

On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
Tito Piezas III's user avatar
6 votes
0 answers
154 views

Fourier transformation of a distribution

We have no idea how to tackle the following Fourier transformation of a distribution: $$ \lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
Y.Okuyama's user avatar
  • 373
6 votes
0 answers
184 views

A class of symmetric functions

When attacking a symmetric problem via an asymmetric method, I encountered the following function: $$U_2(n, m) = \sum_{a = 0}^n\binom na (2^a + 2^{n - a})^m.$$ It is easy to see that this function is ...
WhatsUp's user avatar
  • 3,232
6 votes
0 answers
197 views

Conjecture for a certain Cauchy-type determinant

Given the Cauchy-like matrix $$ \mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{ \Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right) }{ \Gamma(m)\,\Gamma(n) } \frac{m-\frac{3}{4}} {\...
Fred Hucht's user avatar
  • 2,705
6 votes
0 answers
231 views

Closed form for 2D lattice sum

I am wondering if a closed form exists to the lattice sum $$S(a)= \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \frac 1 {(a^2+m^2+n^2)^{3/2}}$$ I am also interested in replacing $m^2+n^2$ with a ...
AndrewBernoff's user avatar
5 votes
2 answers
446 views

Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$

As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...
Milo Moses's user avatar
  • 2,809
5 votes
2 answers
1k views

Integral of a product between two normal distributions and a monomial

The integral of the product of two normal distribution densities can be exactly solved, as shown here for example. I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$): $...
user1172131's user avatar
5 votes
2 answers
804 views

Random walk by simplex vertices

I apologize if this question is well-known, but I was unable to find it mentioned anywhere. There exists a bug which moves around in $r$-space. The bug begins at the origin of this $r$-space. If the ...
Dylan Pizzo's user avatar
5 votes
1 answer
246 views

How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?

I tried to find an inverse Laplace transform by series as follows $$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
Faoler's user avatar
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