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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2
votes
2answers
75 views

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

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
1
vote
0answers
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?
1
vote
1answer
46 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
44 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 ...
3
votes
0answers
67 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 $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
2
votes
1answer
28 views

Constructing an indicator function from a braid group which represents 'all strings have returned to their initial position'.

TL;DR Is there a well-defined closed formula from the braid group $B_n$ to $\{-1,1\} \left(\text{ or }\{0,1\}\right)$ which represents whether all the strings have returned to their initial ...
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
votes
1answer
44 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
2
votes
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} ...
1
vote
1answer
47 views

Identities involving adjoint action

I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that: \begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
3
votes
1answer
53 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
10
votes
1answer
239 views
+50

Packing an infinite sequence of disks

Let $a > 1$ and $Q(a)$ denote the supremum of values of $q$ such that a countably infinite collection of disks, whose areas form an infinitely decreasing geometric progression with the start value ...
2
votes
3answers
178 views

What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$?

A while ago, I started to look at expressions of the following form: $$ S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}, $$ where $p$ is prime, because otherwise things get too complicated for me at ...
0
votes
3answers
46 views

Closed-form term for $\sum_{i=1}^{\infty} i q^i$ [duplicate]

I am interested in the following sum $$\sum_{i=1}^{\infty} i q^i$$ for some $q<1$. Is there a closed-form-term for this? If yes, how does one derive this? I am also interested in ...
2
votes
0answers
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 - ...
1
vote
2answers
47 views

closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$

$$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$ my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function so could some one help ?
1
vote
4answers
72 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
4
votes
1answer
66 views

Closed form of $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$

Recently, I came across the following exercise on the course of discrete math Find a closed form for $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$ So I tried some of the usual techniques: Let ...
0
votes
1answer
38 views

closed form expression for an infinite sum

Is there any closed form expression for the infinite sum $\sum_{n≥0}q^{n^2}u^n$ where both q and n are variables and $n \in N∪0$ ?
0
votes
0answers
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 $n \in N∪0$?
2
votes
0answers
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 $b$, ...
0
votes
2answers
67 views

Is there a closed form solution to this equation?

I have the following equation $e^{-x/b}(a+x) = e^{x/b}(a-x)$ where $b > 0$, and $a > 0$ I need to solve for $x$. I can do it numerically, but would prefer if there was a closed form solution. ...
2
votes
2answers
110 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
5
votes
3answers
170 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
2
votes
1answer
29 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
7
votes
1answer
134 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
15
votes
1answer
198 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
4
votes
1answer
216 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
1
vote
0answers
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 $1/k^2$ ...
0
votes
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 ...
20
votes
4answers
569 views

Evaluate $\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx$

Let $n\ge2$ be an integer , then $$\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx=\text{ ?}, $$ where $\lfloor \space \rfloor$ is the "floor-function"
4
votes
5answers
318 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
18
votes
2answers
278 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
2
votes
0answers
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 ...
1
vote
2answers
45 views

Summation closed form.

I have tried to figure out how to get the close form of: $$\sum_{k=0}^n k^22^{n-k}.$$ I tried to write down each number of the summation but couldn't find any thing to do with that.. can please ...
0
votes
1answer
10 views

How to determine a deficit of base 10 exponent from a minimum unit

I'm trying to determine if a number includes a portion in deficit of a base 10 exponent from a minimum unit. For example, given a minimum unit of $0.01$, $1$ would pass the test, but $1.001$ would ...
5
votes
0answers
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 ...
0
votes
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 ...
8
votes
3answers
202 views

How to evaluate improper integral $\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}dx$?

I'm trying to evaluate the improper integral, $$I(a)=\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}\mathrm{d}x,~~~\text{where }a\in\mathbb{R}^+.$$ Does this integral have a simple closed form ...
3
votes
1answer
71 views

Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
6
votes
2answers
121 views

Superelliptic Area Of $x^5+y^5=r^5$

$${\LARGE\int}_0^\tfrac\pi2\frac{dx}{\bigg(\sqrt[{\Large 5}]{\cos^5x+10\cos^3x\sin^2x+5\cos x\sin^4x}\bigg)^{\large 2}}~=~?$$ Its numerical value is about $1.40171345128228$. Maple, Mathematica, ...
1
vote
1answer
36 views

Does sequences related to function for $lcm(1,2,3 \cdots n)$ exists?

This just came out of curiosity let $$L(n)=lcm(1,2,3 \cdots n)$$ and I know that we can write this with the help of some product involving primes and all . But what I am interested is in Does ...
7
votes
1answer
113 views

Compute $\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$

Does the following series have a closed form ? $$\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$$ Motivation : The original exercise is Compute $\int_{1}^{+\infty} ...
13
votes
2answers
367 views

Evaluating $\int_0^{2\pi}\frac{dt}{\sqrt[4]{P(\cos t,\sin t)}}$

$${\LARGE\int}_0^{2\pi}\frac{dt}{\sqrt[{\LARGE 4}]{A\Big(\sin^8t+\cos^8t\Big)+B\Big(\sin^6t\cos^2t+\sin^2t\cos^6t\Big)+C~\sin^4t\cos^4t}}~=~?$$ where $A=0.3$, $B=-3.3$, and $C=10$. Its numerical ...
0
votes
2answers
85 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
0
votes
2answers
52 views

$x^x-x+5=\frac{29}{4}$

A friend of mine is claiming to have a closed form solution to $x^x-x+5=\frac{29}{4}$, plotting it into wolfram alpha gives an approximation, and the equation doesn't seem very easy to solve. Can any ...
4
votes
2answers
167 views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where ...
3
votes
0answers
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( ...
0
votes
1answer
72 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
0
votes
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 ...