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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378 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 ...
10
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205 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 $$ ...
9
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62 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
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92 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
8
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149 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, ...
8
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134 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 $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
7
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86 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
7
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112 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
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97 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
6
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100 views

Closed-form of $\int_0^{\pi/2} \arctan(x)\cot(x)\,dx$

I'm looking for a closed-form of the following integral problem. $$I = \int_0^{\pi/2} \arctan(x)\cot(x)\,dx.$$ The numerical approximation of $I$ is $$I \approx ...
6
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153 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} ...
6
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147 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). $$ ...
5
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96 views

Infinite series involving factorials of squares

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

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
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126 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
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107 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
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98 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
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71 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
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152 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( ...
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69 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
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51 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 ...
4
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96 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
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127 views

Is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})\ dx$?

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp\left(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a}\right)dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
3
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58 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 ...
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44 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
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75 views

Prove that primitives of $\frac{x^3}{{\rm e}^x - 1}$ have no closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
3
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53 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 ...
3
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35 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
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70 views

Seeking closed-form solution to $\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1}$

I'm looking for a closed-form solution to this infinite series: $$S(\alpha):=\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1},~~~\Re(\alpha)>1.$$ My attempt All I've really been ...
3
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109 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
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49 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
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66 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
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114 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
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106 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
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71 views

How to find $\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$?

I'm trying to calculte the measure of a fat Cantor set, but run into this question: How to find $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
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28 views

Is there a general rule for proving that an equation has no analyticial solution

Somebody asked this here: Prove that an equation has no elementary solution But so far there is no response. The little math I know I have learnt it myself so I dont have a big picture of things. I ...
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49 views

What techniques does Mathematica use to find solutions to these sequences?

This question is related to my previous question: Need help finding a closed form for complicated sum. An answer to that question led my to try and find the general term of the following recurrence: ...
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80 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) ...
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35 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
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38 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
2
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22 views

Closed-form expectation of CES function of a random variable?

I am faced with the following function, called CES (constant elasticity of substitution), of the continuously-distributed random variable $\epsilon$: $f(\epsilon) = (a^\sigma + ...
2
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110 views

Sum of Bessel functions and integral of exponential of sine or cosine

I'm looking to simplify the following sum of Bessel functions of the first kind: $$\sum_{q=-\infty}^{\infty}\frac{(-)^{q}}{2q+1}e^{iq\theta}I_{q}(\alpha^{2})$$ Motivated by a related question and by ...
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49 views

Closed form for the sum $\sum_{a=1}^{b} a^3\cdot (b \bmod a)$

How can we simplify $\sum_{a=1}^{b} a^3\cdot (b \mod a)$? For $a \ge \frac{b+1}{2} $ to $a = b$ it reduces to $$\sum_{a\ge \frac{b+1}{2}}^{b}a^3\cdot (b-a)=b\cdot\sum_{a\ge ...
2
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41 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
2
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92 views

Find the exact value of expression

Let $$S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$$ Is it possible to write $S$ in terms of standard mathematical functions and operators? If yes, what is the exact value of $S$? ...
2
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0answers
71 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
2
votes
0answers
77 views

Closed-form expression of a definite integral

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

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
2
votes
0answers
86 views

Is there a closed form for $\Gamma(i)$?

I know that $$\Gamma(z)\cdot\Gamma(z^*)=|\Gamma(z)|^2\tag{1}$$ and $$\Gamma (z)\cdot \Gamma (1-z) =\frac{\pi }{\sin{\pi z}}\tag{2}$$ but I still can't find closed form for $\Gamma(i)$
2
votes
0answers
75 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...