A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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26
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369 views

Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?

Nested squares seem to be more promising than nested radicals, since they give rational approximations and in principle can be expanded into a series. These two expressions converge numerically: ...
22
votes
0answers
425 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
13
votes
0answers
274 views

A closed form for $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
12
votes
0answers
184 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
12
votes
0answers
164 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\\ ...
12
votes
0answers
549 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
11
votes
0answers
278 views

Closed form of a difficult definite integral

I'm looking for a closed-form expression for the value of this integral: $$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$ The graph of the integrand looks like this: $\hskip 2.4 in$ Numerically, ...
10
votes
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133 views

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Is there a simple expression for the power series $$\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$$ This question came up in a quantum mechanics problem. Mathematica only returns ...
9
votes
0answers
194 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
9
votes
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233 views

Nice way to express the radical $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\dots}}}$

Since $e=\sum_{n=0}^\infty\frac{1}{n!}=1+1+\frac12(1+\frac13(1+\frac14(1+\dots)))$ We have $4^{e-2}=\sqrt{4\cdot\sqrt[3]{4\cdot\sqrt[4]{4\cdots}}}$ Is there however a nice way to express the radical ...
7
votes
0answers
186 views

Known exact values of the $\operatorname{Li}_3$ function

We know some exact values of the trilogarithm $\operatorname{Li}_3$ function. Known real analytic values for $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
6
votes
0answers
251 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
6
votes
0answers
130 views

Infinite series involving factorials of squares

Does $$\sum_{n=0}^\infty \frac{1}{(n^2)!}=2.04167\dots$$ possess a closed form?
6
votes
0answers
191 views

Closed-form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
5
votes
0answers
320 views

Challenging integral: $\int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx$

I'd like to find a symbolic form for the following integral: $$ f(\alpha, Z) = \int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx $$ It is given that $0 \le \alpha \le 1$ and $0 \le Z < 1$. The following ...
5
votes
0answers
154 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
85 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 ...
5
votes
0answers
152 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
128 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
5
votes
0answers
90 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
5
votes
0answers
106 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
5
votes
0answers
163 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( ...
4
votes
0answers
41 views

Closed form solution to an ordinary differential equaiton

How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where $C_1, C_2, C_3, C_4\in \mathbb{R}$ are all constants. It looks ...
4
votes
0answers
55 views

Non-existence of closed-form solutions

An equation like $$a^x+b^x=1$$ can be turned to the form $$t^\alpha+t=1$$ by a suitable change of variable. When $\alpha$ is a rational we can put that in a polynomial form $$u^p+u^q=1$$ and ...
4
votes
0answers
140 views

Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$

I seem to have seen quite a lot of integrals in the form: $$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$ But none of those hold a closed forms (at least to my knowledge) ...
4
votes
0answers
55 views

Can anyone identify the function that represents this infinite product?

$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful ...
4
votes
0answers
155 views

Is there a closed-form expression for Shapley value of glove game?

Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number ...
4
votes
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52 views

How do I find the finite limits of this infinite product?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ I'd like closed form solutions, and in this case that means any ...
4
votes
0answers
177 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
4
votes
0answers
150 views

Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
3
votes
0answers
39 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 ...
3
votes
0answers
46 views
3
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0answers
111 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
3
votes
0answers
49 views

Any closed form for this expression?$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$

I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank! $$ ...
3
votes
0answers
80 views

Infinite Series $\sum_{m=1}^{\infty} \frac{(-1)^{m-1}}{m^m}$

Any ideas to calculate this infinite sum? The ratio test guarantees the convergence; $$\lim_{m\to \infty} \frac{m^m}{(m+1)^{(m+1)}}=0<1$$
3
votes
0answers
119 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
3
votes
0answers
35 views

Does a better form exists for the coefficients of this product of power series?

Let $$f(a,b,t) = \sqrt{1-at}\sqrt{1-bt}$$ We take the series for $\sqrt{1-at}$ and $\sqrt{1-bt}$ around $t = 0$ and multiply them together to find $$f(a,b,t) = \sum_{n=0}^\infty ...
3
votes
0answers
91 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
3
votes
0answers
85 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
3
votes
0answers
43 views

generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

I have been playing around with infinite series and I wondered if it is possible to find an expression for the series: $$ \sum_{k=0}^\infty x^{p(k)} $$ as a generalization of geometric series. $p(k)$ ...
3
votes
0answers
38 views

How to work with a recursive function with 2 recursive instances?

In class, we figured out how to find the closed form of a recursive definition through the "basic 5 steps method". Example function T(n): If n = 1, T(1) = 1 If n > 1, T(n) = T(n-1)+1 Step 1: ...
3
votes
0answers
95 views

Closed-form expression of a definite integral

Does this definite integral admit a closed-form in terms of elementary functions? $$\int_0^{\infty } \frac{x}{\left(x^4+1\right) \left(2 x^2-2 \arctan\left(x^2\right)+\pi \right)} \, dx.$$
3
votes
0answers
52 views

Solving equation with LambertW function?

Does the equation $$ a = b x e^x + c x + d e^x $$ have a solution form solution? I tried to look for it by using the LambertW funcion, but I did not succeed. Thanks in advance.
3
votes
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73 views

Closed form of specific series

I'm working on a problem that involves the integrals of various Bessel functions that Mathematica can't symbolically handle. I've managed to grind out the transformations and integrals by hand, and ...
3
votes
0answers
118 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
3
votes
0answers
108 views

How to resolve this equation for $f(n)$ without using $f(n-1)$

I have an equation related to some work I'm doing on Poisson distribution where I'm calculating a sequence of 100 values between a minimum and maximum value which is set by another formula. ...
2
votes
0answers
50 views

I am looking for a disproof of this conjecture on closed forms

Assume $I=\int_0^\infty f(x)\text{ d}x$ and $J=\sum_{n=0}^\infty f(n) ; I,J\in\Bbb{R}$ Conjecture: If $I$ has a closed form, then $J$ must carry a closed form. Can someone find a proof or ...
2
votes
0answers
28 views

Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when ...
2
votes
0answers
37 views

Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$ \log f(x) + g(x) = c $$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
2
votes
0answers
125 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting: $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...