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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10
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513 views

Are there “mental calculator” persons that can recognize closed-form expressions?

There are people, sometimes called mental calculators, who is capable of performing fast calculations in their head, involving multiplication, factorization, finding logarithms and roots of a high ...
8
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81 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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116 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
6
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135 views

Need help with $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
5
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69 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 ...
4
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69 views

Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,du$

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,du.$$ The parameters are possibly complex, and satisfy $$\Re(c)>-1, ...
3
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68 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
3
votes
0answers
47 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
3
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44 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
3
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70 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( ...
3
votes
0answers
39 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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87 views

Closed form $\int_{a}^b \cfrac{1}{(1+x) \, \left[\ln(1-x)-\ln(1+x)\right]} \, \mathrm{d}x$

I am encountering an integral which involves logarithms, in particular, \begin{equation} \int_{a}^b \cfrac{1}{(1+x) \, \left[\ln(1-x)-\ln(1+x)\right]} \, \mathrm{d}x, \end{equation} where $a$ and $b$ ...
3
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0answers
56 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
108 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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100 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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50 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
98 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
2
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27 views

Closed-form solution to polynomials of this special form?

I have two sets of real numbers $\{a_k\}$ and $\{b_k\}$ such that: $$a_n \leq \ldots \leq a_1 < b_1 \leq \ldots \leq b_n$$ Now consider the following polynomial equation: $$\prod_{k=1}^n (x - ...
2
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56 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
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42 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 ...
2
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64 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
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60 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
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0answers
79 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
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199 views

Integrating a complicated function

After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact): $$y_n(x) = ...
2
votes
0answers
55 views

Converting from Closed Form

Let $A(n) = \lfloor n/2+\log_2(n)-\log_2(2) \rfloor$. Is there an easy way to convert this closed form into a recursive form? If so, what is the general method, and how might it be applied here. If ...
2
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183 views

Double sum with binomial coefficients

Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be ...
2
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0answers
180 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
2
votes
0answers
174 views

What is the closed form of generating function of a power law?

I want to know if there is a "closed form" of the following generating function, $G_n(x) = \sum_{n=0}^{\infty} P_n x^n$ where, $P_n = C(n_0 + n)^{-\gamma}$ where $C$ is a normalization constant, ...
2
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0answers
148 views

Can this series be expressed in closed form, and if so, what is it?

Can this series be expressed in closed form, and if so, what is it? $$ \sum_{n=1}^\infty\frac{1}{9^{n+1}-1} $$
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11 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
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83 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
1
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0answers
59 views

Integral $\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx$

Does the following integral admit a closed-form expression? $$\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx \;\; , \;\; 0 \leq \alpha \leq ...
1
vote
0answers
43 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 ...
1
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0answers
49 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}} ...
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62 views

Not closed under equality?

One of the Peano axioms state that "For any $a \in \Bbb N : a = b, b \in \Bbb N$." An example where transition is not closed under equality is the relation to friends. C may not be B's friend, but A ...
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75 views

Closed-form solution

I wrote down this equation which is the mathematical model of a system. Is there any way to get $V_c(t)$ in a closed-form expression? $V_c(t+d) = V_s \cdot U\left(V_r ...
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59 views

integral of Q-function and modified bessel function

the integral is given in this link http://www.mediafire.com/view/3mpjo9j3stsddz1/Untitled.png where Q(t) is the gaussian error function, I_v(.) is the modified bessel function, anything else is ...
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0answers
75 views

Proof for closed form approximation of e

I am familiar with the derivation of $e$ from a power series, $e = \sum_{k=0}^{\infty} \frac{1}{k!} $ but have not found the proof for the following representation in any textbook $e = \lim_{x\ \to ...
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101 views

Cycle of remainders

Let $N, K, W$ be natural numbers If I start from $R_0$, say any integer $r_0, 0 \lt r_0 \lt N$ and proceed with: $$R_j = ( R_{j-1} + K ) \mod W,\quad j=1,2, \dots$$ (that is the remainder of the ...
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0answers
176 views

Closed form expression for a recurrence relation.

Hello, any ideas for computing closed form for a recurrence relation? In an attempt to compute what the $i$-th post order element would be in terms of its in order position in a complete binary tree, ...
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97 views

How to solve these exponential equations for D?

I'm curious if this is even possible to solve for D. D is the only variable, x, y, z, and w are all constants, and e is the mathematical constant e. $$ (\frac{x+yD^{2}}{zD})^{\sqrt{2D}} = ...
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44 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
0
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0answers
32 views

Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
0
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0answers
11 views

Compact closed form for linear recurrence formulas

Assume you have some linear recursion formula $$f(\vec x)=\sum_{\vec y\in Y}w_{\vec y}f(\vec x - \vec y)$$ Where $\vec y\geq 0 $ and $||\vec y||>0$, $w_{\vec y}\in\mathbb{R}$ and $\vec x , \vec ...
0
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0answers
37 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
0
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0answers
29 views

Simple representation of a Sum

I have a probably pretty simple question. if I have $\sum_{i=0}^{n} 4^i$ and I want a closed representation, Wolfram Alpha gives me: $4^{n+1}/3-1/3$ Why is that? How do I get to that? Thanks!
0
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66 views

Evaluating $\int_2^\infty \zeta(x) - 1 \,\, \mathrm{d}x$

While looking at a table of values for the zeta function, the fact that they approach $1$ made me wonder what the improper integral of the fractional part of the zeta function would be. I've found ...
0
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0answers
58 views

Is there a way to simplify Legendre-squared sum $\sum_{n} \frac{[P'_{n}(x)]^2}{n(n+1)}$

Is there a closed-form expression for $$F(x) = \sum_{n=2,{\rm even}}^{\infty} \frac{[P'_{n}(x)]^2}{n(n+1)}$$ where $P'_{n}(x)$ is the derivative of the $n^{\rm th}$ Legendre polynomial? Simple ...
0
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0answers
80 views

Proof of a striking identity of Tito Piezas III

In the q series blog of Tito Piezas here . He gives a very striking relation I am wondering on how to prove that ?
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0answers
32 views

Help with formulating a mathematical logic formula

I need to write a precise mathematical expression to formulate an algorithm that could be implemented in software. It has the following simple logic: An Internet user of the software in a ...