43
votes

Accepted

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

Following the suggestion I made in a comment, the integral can be rewritten as the contour integral
$$
I_{3,2} = \frac{1}{2\pi i} \oint \frac{\operatorname{tanh}^3 z}{z^2} \log(-z) \, dz ,
$$
where ...

39
votes

Accepted

### Why these surprising proportionalities of integrals involving odd zeta values?

For $n\geq 1$ and $m\geq 0$, an application of integration by parts ($u=\log^n(1-x)$, $dv=\log^m(x)\,dx/x$) followed by the substitution $x\mapsto 1-x$ shows that
$$
\frac{I_{n,m}}{I_{m+1,n-1}}=\frac{...

36
votes

Accepted

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

There is no such function. Since $f$ would have to map $\mathbb R$ onto $\mathbb R$ for the equation to make sense at all $x\in\mathbb R$, it follows that $f^{-1}(x)\to -\infty$ also as $x\to -\infty$,...

30
votes

### When can an invertible function be inverted in closed form?

I recommend the following paper:
MR1501299
Ritt, J. F.
Elementary functions and their inverses.
Trans. Amer. Math. Soc. 27 (1925), no. 1, 68–90.
(freely available on the web). It indeed gives a short ...

28
votes

Accepted

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

Consider the integral $$I=\int_0^1\int_0^1\frac{zdzdt}{(1-zt)(1-z(1-t))}=\sum_{k,j\geqslant 0} \int_0^1\int_0^1 z^{k+j+1}t^k(1-t)^jdzdt=\\
\sum_{k,j\geqslant 0} \frac1{k+j+2}\cdot \frac{k!j!}{(k+j+1)!}...

20
votes

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

Rewrite the integrand and apply Taylor expansion to $\frac1{(1+e^{-2x})^3}$ so that
$$\frac{\tanh^3x}{x^2}=\sum_{j\geq0}(-1)^j\binom{j+2}2
\frac{(1-e^{-2x})^3}{x^2}\binom{j+2}2e^{-2jx}.$$
Integrate ...

18
votes

Accepted

### On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function

First, $\mathrm{gd}^{-1}(z) = \ln \frac{1 + \tan(\frac{z}{2})}{1 + \tan(\frac{z}{2})}$. So we substitute $t = \tan(\frac{z}{2})$, obtaining:
$$\mathcal{G}_n = \int\limits_{0}^{1}2\cdot\ln^{n + 1}\...

16
votes

Accepted

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

Let me first establish the case $m=2$. The general case is established in the "Added" section below.
By making the substitution $x=\sinh t$ we get
\begin{align*}
I(n,2)&=\int_0^\infty t^n\frac{\...

16
votes

### Is there real or complex analytic function whose positive real zeros are the primes?

H. Laurent introduced the function
$$f(z)=\sum_{n=1}^\infty(\sin\pi z)^2\left(\frac{1}{n^2\sin\frac{\pi z}{n}}-\frac{1}{\pi n (n-z)}\right)^2$$
whose real roots are the positive primes and have no ...

15
votes

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

I'm not quite sure what you mean by an "analytic formula." As Fedor Petrov indicated, there is unlikely to be a closed formula. However, there is a convergent power series. More precisely, consider ...

14
votes

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

This is a second answer with a somewhat different viewpoint from my other answer. It is an expansion, in some sense, of Gerry Myerson's answer. There is a general theory for estimating these sorts of ...

13
votes

Accepted

### Invertibility of specific function

The answer to the question of whether the inverse has a closed form depends of course on one's definition of "closed form." One plausible definition is that a closed-form function is a ...

Community wiki

13
votes

Accepted

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

Fiddling with Maple, I get: if $n$ is a positive real number, then
$$
{{_4\mathrm F_3}(n,n,n,2\,n;\,n+1,n+1,n+1;\,-1)}={\frac {{n}^{2}
\sqrt {\pi}\,\Gamma(n+1)\,\psi^{(1)}(n)}{{4}^{n}\,\Gamma \left( n
...

13
votes

Accepted

### 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"

We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and
$$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you
call Gieseking's constant but which is simply the value at 2 of the
L ...

12
votes

Accepted

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

Some experimentation suggests that the second series is given by
$$
S_4^-(a) = \sum_{n=-2a}^\infty (n+a) \binom{2a}{-n}^4 = \frac{1}{4\cos(2\pi a) \,\Gamma(2a+1)^2\,\Gamma(-4a)},
$$
which agrees with ...

11
votes

### When can an invertible function be inverted in closed form?

Closed-form functions need the definition which set of functions is allowed to represent the function. Take e.g. the algebraic definition of the Elementary functions by Liouville and Ritt (Wikipedia: ...

11
votes

Accepted

### On Zagier's missing continued fraction with multiple limits?

Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and
$P=2\pi^2/81$. The limits are almost certainly (not proved),
\begin{align}
\lim_{m\to\infty}C_2(6m+0) &= -Q\\
\lim_{m\to\infty}...

10
votes

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

The sequence (at any rate, the case $q_0=1$) has been studied, and references are given at OEIS. The closest thing to a formula given there is $a(n) = [c^{2^n}]$ for $n > 0$, where $c = 1....

10
votes

Accepted

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

This is the Fourier series for the RHS, as a function of $\alpha\in (0,2\pi)$,
$$
f(\alpha)=\frac{2\pi i}{1-e^{-2\pi iz}}\, e^{-iz\alpha} .
$$
The series representation follows by computing the ...

9
votes

### Are these two new ways of representing odd zeta values as integrals known?

Using the reflection formula followed by the recurrence formula and the Beta integral representation (DLMF)
\begin{align}
\frac{x(1-x)}{\sin \pi x}&=\frac{1}{\pi}x(1-x)\Gamma(x)\Gamma(1-x)\\
&...

9
votes

### counting points on unit sphere mod p

Since this old question has resurfaced, let me sketch two ways to prove the stated formulæ using algebraic geometry:
The first way is fairly elementary. Let us stick for definiteness with the number ...

9
votes

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

Yes, some special cases of the hypergeometric function give roots of
polynomial equations whose Galois groups are not solvable.
The simplest examples are the solutions of $y(1-y)^t = x$ for rational $...

9
votes

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

If we denote $A_n=q_n+1/2$, then
$$A_n=A_{n-1}^2+5/4$$
with $A_0=q_0+1/2\ge 3/2$ by $q_0\in\mathbb{N}$.
Further,
$$\log A_n=2\log A_{n-1}+\log\left(1+\frac{5}{4A_{n-1}^2}\right),$$
namely
$$\frac{1}{...

9
votes

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

A summation method for this...
$$
F(s) = -\sum_{n=2}^\infty \frac{(-1)^n}{n^s} = -(2^{1-s}-1)\zeta(s)
\qquad\text{for $s>0$}
$$
Differentiate:
$$
\sum_{n=2}^\infty \frac{(-1)^n\log n}{n^s} = F'(s)
...

9
votes

Accepted

### Partition numbers as the specific sums of the A161511

If $2m=2^{j_1}+2^{j_2}+\ldots+2^{j_k}$ for $0<j_1<j_2<\ldots<j_k$, then $\ell(m)=j_k-1$, $a(m)=j_1+(j_2-1)+(j_3-2)+\ldots+(j_k-(k-1))$, as the $i$-th (from the left) 1 in the binary ...

8
votes

Accepted

### Does this system have a closed-form solution? $x_j^2 = \sum_{i=1}^n B_{ij} x_i$

Sorry, there cannot be a simple solution. For example,
taking $n=3$ and $B_{ij} = (i+j-1)/6$, we compute numerically
(by iterating the contraction mapping as you suggest)
$$
(x_1,x_2,x_3) = (1....

8
votes

### On the search for an explicit form of a particular integral

Let
$$
I_n=\int_0^1 \frac{x^n x^x (1-x)^{1-x} \sin (\pi x)}{\pi e}\mathrm dx
$$
and
$$
b_0=-1,\quad b_n=\frac{-1}{n}\sum_{k=1}^n\frac{b_{n-k}}{k+1}
$$
Then
$$
\boxed{I_n\equiv b_{n+2}}
$$
so indeed ...

8
votes

Accepted

### Difficult trigonometric integral

Here is an outline of the approach I have taken to solve this integral.
First rewrite the integral $(1)$ in Cartesian variables:
$$I=\int_{-\infty}^{\infty} \mathrm{d}^3v~ \frac{3x^2y^2v_z^2}{y^...

8
votes

### how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$

There always exists at least one solution: If $x >\mu_i$ for all $i$, then each of the terms in the sum is positive and if $x < \mu_i$ for all $i$, then each of the terms in the sum is negative. ...

8
votes

### Invertibility of specific function

The change of variable given above by Pietro Majer shows that this is equivalent to Kepler's equation Wikepedia on Kepler's Equation which is believed not to have any closed form solution (let alone ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

closed-form-expressions × 209sequences-and-series × 57

nt.number-theory × 43

integration × 42

ca.classical-analysis-and-odes × 29

co.combinatorics × 27

special-functions × 22

recurrences × 20

pr.probability × 16

polynomials × 10

hypergeometric-functions × 10

probability-distributions × 9

reference-request × 8

real-analysis × 8

riemann-zeta-function × 8

binomial-coefficients × 8

gm.general-mathematics × 8

integer-sequences × 7

gamma-function × 7

cv.complex-variables × 6

analytic-number-theory × 6

st.statistics × 6

galois-theory × 6

power-series × 5

expectation × 5