A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

learn more… | top users | synonyms

1
vote
0answers
50 views

Packing an infinite sequence of disks

Let $r > 1$ and $Q(r)$ denote the supremum of values of $q$ such that an infinite sequence of disks, whose radii form an infinitely decreasing geometric progression with the start value $1$ and the ...
0
votes
3answers
139 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
44 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
26 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
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. ...
13
votes
2answers
357 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 ...
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 ?
4
votes
1answer
65 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 ...
9
votes
2answers
196 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
2
votes
0answers
53 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 $b$, ...
27
votes
3answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
18
votes
2answers
272 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\, ...
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. ...
20
votes
4answers
567 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"
34
votes
3answers
654 views

$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

I need to find a closed-form for the following integral. Please give me some ideas how to approach it: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
2answers
97 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 ...
3
votes
1answer
113 views

An example of Risch algorithm for integrating $y$, $F(x,y)=0$.

I would like to compute the integral $$ \int y\,dx, \qquad y=\sqrt{x+\sqrt{x+1}},\\ F(x,y)=y^4-2xy^2+x^2-x-1=0, $$ in closed form, where $F(x,y)$ is a polynomial in $\mathbb{C}[x,y]$. I am trying to ...
1
vote
1answer
248 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: M is a non-empty set in an Inner Product Space (IPS) X. I need to show that the annihilator of M which is ...
13
votes
3answers
296 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
5
votes
3answers
165 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\\ ...
15
votes
1answer
190 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 ...
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? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
5
votes
1answer
101 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: ...
4
votes
1answer
197 views

Expressing the solutions of the equation $ \tan(x) = x $ in closed form.

I know that the equation $ \tan(x) = x $ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t ...
18
votes
1answer
393 views

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Is it possible to simplify the ratio $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
12
votes
2answers
127 views

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

I need to evaluate the following integral with a high precision: $$ I=\int_{0}^{\infty}\left[% {\pi^{2} \over 6} - {\rm Li}_2\left({\rm e}^{-x}\right) -{\rm Li}_2\left({\rm ...
4
votes
1answer
208 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
82 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 $1/k^2$ ...
4
votes
5answers
316 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 ...
0
votes
0answers
30 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 ...
12
votes
1answer
182 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
9
votes
3answers
177 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
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 ...
38
votes
1answer
1k views

Conjecture $\int_0^1\frac{dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. Can we ...
1
vote
2answers
44 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 ...
5
votes
0answers
62 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
votes
2answers
114 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, ...
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 ...
8
votes
3answers
192 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 ...
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 ...
3
votes
1answer
69 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 ...
1
vote
4answers
106 views

How to solve a recursive equation

I have been given a task to solve the following recursive equation \begin{align*} a_1&=-2\\ a_2&= 12\\ a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2, \quad n \geq 3. \end{align*} Should I start by ...
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 ...
32
votes
1answer
937 views

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
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} ...
0
votes
2answers
84 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.
4
votes
2answers
165 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 ...
1
vote
1answer
63 views

May I know that there is a special structure or solution on this linear fractional optimization?

I am not familar with the optimization problems, but I want to know a very simple formulation: \begin{array}{cc} {{\max_\mathbf{x}}} & \frac{\mathbf{a}^{T}\mathbf{x}}{\mathbf{b}^{T}\mathbf{x}}\\ ...
2
votes
1answer
72 views

A closed form for $\sum_{i\cdot j^k=n}(-1)^i$?

$$\alpha_k(n) \stackrel{\text{def.}}{=} \sum_{i\cdot j^k=n}(-1)^i.$$ Does a closed form exist for $\alpha_k(n)$? For low values of $k$: $$\alpha_0(n)=(-1)^n$$ $$\alpha_1(n)=\begin{cases} ...
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 ...