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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6
votes
3answers
97 views

what's the summation of this finite sequence?

$a$ and $b$ are positive integers. The summation is $$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}} {a + b - x}\\ b \end{array}} \right)} .$$ Any closed-form expression? I thought it ...
5
votes
1answer
212 views

Closed-form formula for the $n^{\rm th}$ term of ${1,1,1,1,\ldots, 1}, {2,2,2,2,\ldots, 2},\ldots, {k-1, k-1}, k.$

Let $k$ be a positive integer. Consider a finite sequence $L_k(n)$ given by $$\underbrace{1,1,1,1,\ldots, 1}_{k\text{ terms}}, \underbrace{2,2,2,2,\ldots, 2}_{k-1\text{ terms}},\ldots, ...
5
votes
1answer
125 views

General Solution of $y'(x)+p(x)e^{r(x) y(x)}=q(x)$

I solved the case for the non-homogenous constant coefficients case and I wondered if there is a way to find a general solution to a non-constant coefficient case. I don't know how to approach this at ...
1
vote
1answer
52 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
3
votes
2answers
146 views

Solving a 2 independent variables (2nd degree) recurrence relation

Changes to the recurrences and definition are changed! See here: $f(n, 1) = 2n^2 $ and $f (n, k) = 0$ for $k \geq 2n$ and for $k < 0$ and $f(n, 2n-1) = 1$ for all $n$. Question: Is it possible ...
7
votes
2answers
151 views

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

How to find this integral closed form: $$I=\int_{0}^{+\infty}\operatorname{sech}^2{(x^2)}\,dx$$ where $\operatorname{sech}{(x)}$ is defined as secant of hyperbolic function. This problem ...
1
vote
0answers
40 views

New identity for sums of Bessel functions?

I've come across the following proposed identity: $$ ...
2
votes
3answers
46 views

Closed form for certain trigonometric integral

\begin{align}&\mbox{Is there a closed form for} \\[2mm]&\int_0^{\pi/2} \sin^{2}\left(\, nx\,\right)\sin\left(\, mx\,\right)\cot\left(\, x\,\right) \,{\rm d}x\ \quad\mbox{where}\quad m, n\ ...
0
votes
0answers
29 views

List of functions $\chi_{s,a}(n)$ defined on a Group such that $\chi_{s,a}(n)\in{s,a}$ and depending on the parity

Question Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property $$\chi_{s,a}(n)= \begin{cases} s, ...
3
votes
0answers
42 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
2answers
61 views

An infinite exponential sum

I was trying to create a problem for a test I'm writing, and I ended up attempting to evaluate $$\sum_{n=1}^{\infty} \dfrac{1}{e^n-1}.$$ This definitely converges, but I have no idea how to go ...
1
vote
0answers
40 views

Zeros of $f$ in a disk

If $f$ holomorphic in a domain $U$ and $f(z)\neq 0$ for all $z\in U$ then every zero of $f$ is such that $f(q)=0$ and $\det(Df_{p})>0$. Using that I have to prove that if $f$ keeps that conditions ...
0
votes
1answer
49 views

How to solve integrals using series?

Many places I have seen when solving integrals you change a lot of it into sums. Finding $\int_{0}^{\pi/2} \dfrac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$ Is just an example. So in general, how do you ...
0
votes
1answer
79 views

$\displaystyle\int_0^{\infty} \dfrac{\mathrm{d}x}{(x^4+ax^3+bx^2+cx+d)^m}$

It's a generalization for $$F(a,m)=\int_0^{\infty}\dfrac{\mathrm{d}x}{(x^4+2ax^2+1)^{m+1}}$$ that's being evaluated at Irresistible integrals, George Boros and Victor H. Moll 2004. I wonder if there ...
27
votes
3answers
839 views

Closed form of $\displaystyle\mathscr{R}=\int_0^{\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx$

Inspired by Mr. Olivier Oloa in this question. Does the following integral admit a closed form? \begin{align} \mathscr{R}=\int_0^{\Large\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx ...
6
votes
3answers
260 views

The Integral of Multiple Tangent Functions

I need help to find the numerical values to the precision at least $50$ digits (the closed forms if possible) for the following integrals \begin{equation} ...
2
votes
1answer
99 views

summation of a finite sequence?

What is the summation of the finite sequence: $$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}} {2i - 2}\\ {i - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {2n + 2 - 2i}\\ ...
18
votes
0answers
428 views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
5
votes
1answer
121 views

what is the summation of such a finite sequence?

The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$ The answer is $4^n$. How to prove it, and how to think out it?
9
votes
2answers
121 views

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

Can the following double integral be evaluated analytically \begin{equation} I=\int_{0}^{\Large\frac{\pi}{4}}\int_{\Large\frac{\pi}{2}}^{\large\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin ...
17
votes
2answers
245 views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ ...
5
votes
0answers
116 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 ...
9
votes
2answers
96 views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

$\def\Li{\operatorname{Li}}$ I wonder how to prove: $$ \int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $$ I'm not used to polylogarithm, so I don't know how to tackle it. ...
15
votes
5answers
212 views

Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$

I would appreciate to understand the main steps giving the evaluation of this series: $$ S=\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$$ where $H_n$ is the harmonic number. I've tried with no ...
14
votes
1answer
164 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

$\def\Li{{\rm{Li}}}$I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose ...
5
votes
1answer
61 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
3
votes
2answers
97 views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

I have got a trouble with integral $$\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \, \mathrm{d}x,\quad |\alpha|<1.$$ My teacher said there are two ways of solving such ones, if there is ...
1
vote
3answers
31 views

Closed form of a sum

I am trying to derive the closed form of the sum $\sum\limits_{i=2}^n \frac{1}{i(i-1)}$ which Mathematica tells me is $\frac{n-1}{n}$. I am completely baffled on how to arrive at this result. The ...
9
votes
0answers
53 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 ...
4
votes
0answers
35 views

Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form

$\underline{Given:}$ Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$ using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$ $\underline{My\ Work:}$ I substituted ...
6
votes
1answer
142 views

Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions

I think the following identity is true. How could we prove it? $${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + ...
5
votes
0answers
98 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 ...
1
vote
0answers
21 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. $p(k)$ ...
0
votes
4answers
40 views

why is $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$ [closed]

how do I get that $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$? And why is it true?
7
votes
1answer
126 views

How to calculate the closed form of the series

We know that the closed form of the series $$\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k=-\frac{11\pi^4}{360}+\frac{\ln^42-\pi^2\ln^22}{12}+2\mathrm{Li}_4\left(\frac12\right)+\frac{7\ln 2}{4}\zeta(3),$$ but ...
5
votes
1answer
60 views

Three integral involving polylogarithm function

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 ...
7
votes
0answers
65 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 $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
5
votes
0answers
84 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 $c_1 = \operatorname{Li}_2(i)$, ...
11
votes
3answers
171 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
12
votes
4answers
511 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
15
votes
6answers
329 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
5
votes
2answers
148 views

Closed formula for the asymptotic limit of a definite integral

I would like to solve the following integral: $$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$ in the limit where $b$ is small ($a$ and $b$ are positive constants). ...
3
votes
2answers
58 views

Decomposition of $_1F_2(1+n;1,2+n;x)$

I am looking for a way to decompose $_1F_2(1+n;1,2+n;z)$ for $n\in\mathbb{N}$ into either Bessel J functions or regularized confluent hypergeometric functions $_0\tilde F_1(b(n),z)$. Mathematica seems ...
6
votes
3answers
102 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
7
votes
1answer
172 views

Closed-form of $\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx$

We know that $$\int_0^1 \int_0^1 x^y\,dy\,dx = \ln 2.$$ Do we know a closed-form of $$\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx\,?$$ As a start we know that $$\int_0^1 x^{(y^z)}\,dz = ...
2
votes
2answers
170 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
1
vote
1answer
104 views

A inverse Trigonometric multiple Integrals

How to calculate the closed form of the integral $$\int\limits_0^1 {\frac{{\int\limits_0^x {{{\left( {\arctan t} \right)}^2}dt} }}{{x\left( {1 + {x^2}} \right)}}} dx$$
4
votes
1answer
75 views

How to prove $H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2}$

How to prove: $$ H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2} $$ Where $C$ is Catalan's number, $A$ is Glaisher-Kinkelin's constant and $H(x)$ is the ...
3
votes
1answer
139 views

Closed-form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
8
votes
1answer
141 views

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...