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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3
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0answers
54 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 $$I(a)=\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
4
votes
2answers
88 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
0
votes
1answer
25 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
13
votes
2answers
83 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
4
votes
4answers
72 views

How to derive the closed form of the sum of $kr^k$

$$ \sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $$ How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know ...
6
votes
2answers
105 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
2
votes
1answer
48 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
0
votes
0answers
18 views

Closed-form solution of the following LP problem

I am considering the following LP problem: $$ \begin{array}{cl} \text{maximize} & c^Tx\\ \text{subject to} & a^Tx\geq0 \\ & 0\leq x\leq x^\max \end{array} $$ where ...
0
votes
2answers
27 views

Why is this term $=1$

Can you tell me why $$\frac{1}{r} \sum_{k=0}^{r-1} R_N(x^k) \sum_{s=0}^{r-1} e^{\frac{-2 \pi i s k}{r}}=1?$$ Here $R_N(x^k)$ is the remainder of $x^k$ Modulo $N$. When I entered the last sum in ...
2
votes
0answers
31 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 ...
1
vote
3answers
42 views

How to find a closed form of this simple factorial sequence [duplicate]

$$S_1=1\\ S_n=n!+S_{n-1}$$ Is there a simple way to express $S_n$ without summing up all the previous terms? Sorry I haven't put any effort in the problem but I don't know where to start. So this ...
13
votes
4answers
288 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\sum_{k=1}^nk^{-1}$ is a harmonic number. Could you help me with it?
4
votes
1answer
56 views

Can you prove a definite integral has no closed form?

It is a well known fact that some functions posses no closed form antiderivative yet still they have definite integrals that have a closed form. A classic example is the Gaussian integral ...
11
votes
5answers
725 views

How to deduce a closed formula given an equivalent recursive one?

I know how to prove that a closed formula is equivalent to a recursive one with induction, but what about ways of deducing the closed form initially? For example: $$ f(n) = 2 f(n-1) + 1 $$ I know ...
32
votes
5answers
759 views
+200

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
8
votes
2answers
163 views

Evaluation of a dilogarithmic integral

Problem. Prove that the following dilogarithmic integral has the indicated value: $$\int_{0}^{1}\mathrm{d}x ...
2
votes
2answers
74 views

Solution of an equation involving even integers

If $x$ is any positive even integer $> 1$. I compute: $$ c = x + x! $$ Now assume instead $c$ (even integer) is given, and I want to get back the value $x$. Is it possible to find a simple ...
20
votes
4answers
455 views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
3
votes
2answers
43 views

A sum of difference of floors

I have the sum ( $M$ is any integer $> 1$ ): $$ \sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor -\left\lfloor\, 2M \over h\,\right\rfloor\,\right) $$ and looking for a way to ...
3
votes
0answers
50 views

Evaluating the integral $\int \frac{dx}{\sqrt{a(1+x)^3+1}}$ [duplicate]

How can I solve this integral? $$\int \frac{dx}{\sqrt{a(1+x)^3+1}}$$ where $a>0$. I have tried by using Mathematica, but it fails. Someone has any sugestion?
5
votes
1answer
89 views

Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
5
votes
1answer
109 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
4
votes
1answer
78 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
1
vote
1answer
31 views

How do I proceed from here on finding the Binet's formula via generating functions?

So, I'm stuck with the algebra for the nth number on the Fibonacci sequence in here. I managed to get to the part where $G(x) = \frac{x}{1-x-x^2}$ $=$ $\frac{x}{(1-\alpha x)(1-\beta x)}$, and I know ...
1
vote
0answers
115 views

The closed form of $\sum_{n=1}^{x}n!$

Let $$y=\sum_{n=1}^{x}n!$$ be the sum of consecutive factorials. What is closed form for $y$ in terms of $x$? Wolfram Alpha says that $$y=-(-1)^x\Gamma(x+2)(!(-x-2))-!(-1)-1$$ where $!x$ is ...
16
votes
2answers
445 views

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
2
votes
2answers
71 views

Sum of floor of ratios

I need to compute, in a program at work, the sum, for $k = 2$ to $n-1$, of the floors of the ratios: $\frac{n}{k}$. Since n is a large integer in my case I would need a "closed form" formula for this ...
7
votes
1answer
74 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
0
votes
2answers
44 views

Closed form for recurrence relation

Is there a closed-form solution to the following recurrence: $$T(n) = T(n-1) + T(n-3)$$ If yes, what is it and how can it be proven/derived? If not, then why because a somewhat similar recurrence ...
8
votes
0answers
98 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 $$ ...
2
votes
1answer
64 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
2
votes
1answer
98 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
4
votes
1answer
174 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
10
votes
0answers
168 views
6
votes
0answers
119 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
votes
1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
16
votes
2answers
335 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
3
votes
1answer
99 views

Evaluating an infinite square root

How do I evaluate the square root: $$\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\cdots}}}$$ I have tried creating two arithmetic sequences such that $$a_n = 1999+14n$$ $$b_n = 274+2n$$ so the square ...
11
votes
2answers
370 views

Evaluating $\int_0^\pi\arctan\left(\frac{\ln\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
2
votes
2answers
44 views

Sum of odd Bessel Functions

In an answer, I showed that: $$\sin(1)=2\sum_{k=0}^\infty(-1)^k J_{2k+1}(1)$$ Where $J_n(x)$ is the Bessel function of the first kind. Is there a more general result for the infinite sum of odd Bessel ...
21
votes
3answers
527 views
+500

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

Please help me to evaluate this integral: $$ I={\large\int}_{0}^{\infty}{\ln\left(x\right) \over 1 + x}\, \,\sqrt{\,x + \sqrt{\,1 + x^{2}\,}\, \over 1 + x^{2}\,}\,\,{\rm d}x.\tag1 $$ Mathematica could ...
2
votes
0answers
63 views

BCH (Baker-Campbell-Hausdorff) formula for $[X,Y]=xY-yX$

If some $X$ and $Y$ satisfy the commutation relation $[X,Y]=XY-YX=xY-yX$, where $x$ and $y$ are numbers (or commute mutually and with $X$ and $Y$), then what is the closed form of $\ln(\exp X \exp ...
0
votes
1answer
39 views

Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case?

The question was motivated by the way in which we approach the convergence and divergence of some series. During my undergraduate analysis course one of the only times in which the partial sum was ...
2
votes
0answers
62 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 $S$? ...
6
votes
1answer
120 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
4
votes
2answers
157 views

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
10
votes
2answers
302 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
7
votes
1answer
100 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?
1
vote
1answer
47 views

Closed form for the recursion $\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$

I was completing a computer science problem when the following recursion popped up: $u_0=1$ $\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$ Is there a closed form for this recursion ? I ...
1
vote
2answers
54 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...