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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4
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1answer
179 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
0
votes
2answers
87 views

Generating function and its closed form

Consider the inequality $x_1 + x_2 + x_3 + x_4 ≤n$ where $x_1,x_2,x_3,x_4,n ≥ 0$ are all integers. Suppose also that $x_2 ≥ 2$, that $x_3$ is a multiple of 4, and $1 ≤ x_4 ≤ 3$. Let $c_n$ be the ...
0
votes
1answer
44 views

Does this sum of products of binomial coefficients have a simple closed form?

Let $c,m,k$ be positive integers. Is there a simple closed form for the following sum? $$ \sum_{i=1}^{c-1} (-1)^i {c \choose i} {im \choose k} $$ Mathematica finds nothing, and Maxima's implementation ...
1
vote
0answers
66 views

Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...
6
votes
2answers
198 views

Closed form of the sum $\sum\limits_{n=0}^\infty \exp(-n^3)$

I am trying to calculate the sum of the series $$\sum_{n=0}^\infty \exp(-n^3)$$ Can it be expressed in terms of known mathematical functions?
0
votes
0answers
19 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
0
votes
2answers
78 views

Closed form for nth term - generating functions

I think I am mostly confused about what the question is asking. I read that "closed form" means that it should not be represented as as infinite sum, so I am not sure what they are asking for. Would ...
2
votes
2answers
100 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...
2
votes
1answer
78 views

How to solve the following equation? $\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$

I have no clue: $$\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$$
1
vote
0answers
29 views

Closed form for $\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$

Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation) If so, how can we obtain it ? If there isn't a closed ...
1
vote
2answers
95 views

Solving 2nd order linear recurrence with non-constant coefficients

I am trying to find a general solution to the following definite integral: $$F_{n}{\left(a,b;z\right)}:=\int_{a}^{z}\frac{x^{n}}{\sqrt{\left(x-a\right)\left(b-x\right)}}\,\mathrm{d}x,\tag{1}$$ ...
1
vote
2answers
56 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
1
vote
0answers
68 views

Closed form of generating function $r^a$

Find the closed form of the generating function of $r^a$. in this question $r$ is the variable part and $r$ assumes the values $1,2,3,4,5,6,7 \ldots$ and $a \in \mathbb R_{\geq 0}$. I would appreciate ...
4
votes
1answer
101 views

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by ...
0
votes
1answer
30 views

Find closed form of recursion

I know how to get the equation of the form $x^2 = Ax + B$ and then from there get $a_k = C * x_1^k + D * x_2^k$ but doesn't the original $b_k$ equation have to be of the form $7b_{k-1} - 10b_{k-2}$ ...
2
votes
0answers
77 views

How to find $\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$?

I'm trying to calculte the measure of a fat Cantor set, but run into this question: How to find $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
0
votes
1answer
51 views

The $C_0-$group generated by the operator $(Af)(x)=f'(x)+a(x)f(x)$

Consider the Banach space $L^1(\mathbb{R})$ of integrable functions $f:\mathbb{R}\to \mathbb{R}$. Consider the unbounded operator $A$ defined by $$(Af)(x)=f'(x)+a(x)f(x), \ \ \ x\in \mathbb{R}$$ for ...
0
votes
1answer
63 views

Expressing an integral in closed form

Is there a closed-form expression for this integral? $$\int \frac{\sin(Ax/2)}{A\sin(x/2)}\mathrm{d}x$$
0
votes
1answer
59 views

A Trig Integral

Does the integral \begin{align} \int_{0}^{\pi/2} \cos(x) \, \ln\left( \frac{1 + a^{2} \sin(x)}{1 - a^{2} \sin(x)} \right) \, dx \end{align} have a closed form and what is changed if the limits are ...
3
votes
3answers
159 views

How to solve:$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
2
votes
4answers
182 views

Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
11
votes
2answers
191 views

Closed form for ${\large\int}_0^\infty\frac{x\,\sqrt{e^x-1}}{1-2\cosh x}\,dx$

I was able to calculate $$\int_0^\infty\frac{\sqrt{e^x-1}}{1-2\cosh x}\,dx=-\frac\pi{\sqrt3}.$$ It turns out the integrand even has an elementary antiderivative (see here). Now I'm interested in a ...
6
votes
3answers
137 views

A closed form for the sum of $(e-(1+1/n)^n)$ over $n$

I have been having some trouble trying to find a closed form for this sum. It seems to converge really slowly
9
votes
1answer
151 views

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

How to find the value of $$I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$$ If we put $$I_2=\int_0^\infty\frac{\arctan^2({x})\log({1+x^2})}{\sqrt{x}(1+x^2)}dx$$ After long ...
0
votes
1answer
66 views

Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
1
vote
1answer
86 views

Transcendental equations involving more than 2 terms

I now how to solve transcendental equations involving only two terms like: $xe^x=k$ $x=W(k)$ Where W(x) is the Lambert's Omega function. But how can I solve (for $x$) a more general case? Like: ...
1
vote
2answers
55 views

Calculating closed forms of integrals

So I've been told that you can't find the closed form of $\int e^{-\frac{x^2}{2}}$. Apparently, you know the exact result then you integrate over the whole of $\mathbb{R}$ but every other number ...
1
vote
0answers
67 views

Can $\int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2-x^2})}e^{ikx}dx$ be found in closed form?

I am trying to see if it is worth pursuing to try to calculate the following integral analytically: \begin{align} \int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2 ...
9
votes
5answers
215 views

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

I have found myself faced with evaluating the following integral: $$\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx. $$ Mathematica gives a closed form of $8 \pi ^3/(81 \sqrt{3})$, but I have no ...
0
votes
1answer
30 views

Get a closed form of an expression

I try to get a closed form of the following function $f(x)$. $a_0\left(x\right)=x$ $a_{n+1}\left(x\right) = x^{a_n\left(x\right)}$ e.g. $a_{3}\left(x\right) = x^{ \left( x^{ \left( x^x \right) } ...
6
votes
3answers
124 views

Any given function $f\colon [0,1] \to \Bbb R$, what is $\int_0^1 \frac{f(x)}{f(x)+f(1-x)} \, dx$?

I have a general function $$\int_0^1\frac{f(x)}{f(x)+f(1-x)}dx.$$ How do I solve it? I have tried to split it up from $0$ to $0.5$ and from $0.5$ to $1$, but I don't know what to do next. Thanks for ...
1
vote
0answers
45 views

How to manipulate this summation in the easiest way possible?

$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f ...
1
vote
1answer
47 views

Does this series have a closed form?

A friend of mine asked me if I could find a closed form for the series: $$ S = \sum_{n=-\infty}^{\infty} (n-h)^{\alpha} e^{-\beta(n-h)^2}, $$ with $\alpha,\beta > 0$. I don't even know how to ...
3
votes
0answers
55 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
0
votes
1answer
64 views

Sequence closed expression or others

What are closed expression or any other expression (involving integrals, specials functions...) for $\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$ where $t>0$ integer Thank you
6
votes
0answers
110 views

Closed-form of $\int_0^{\pi/2} \arctan(x)\cot(x)\,dx$

I'm looking for a closed-form of the following integral problem. $$I = \int_0^{\pi/2} \arctan(x)\cot(x)\,dx.$$ The numerical approximation of $I$ is $$I \approx ...
0
votes
1answer
27 views

Closed form expression for $\sigma$

A student I'm tutoring came to me with a problem in which he needs to find a closed-form expression in $\sigma$, $E(|Y|)$. $Y$ has a normal distribution with mean $0$ and standard deviation $\sigma$. ...
0
votes
1answer
50 views

Closed form for this 2 variable recurrence?

I'm trying to find a closed form for this two variable recurrence, but Wolfram Alpha does not seem to understand the input. $$ \begin{cases} a_{0,1} = 1 \\ a_{0,i} = 0 \quad \forall i\neq1 \\ ...
0
votes
1answer
73 views

Manipulation of summations

this question branches off another question that can be seen here Now we begin be taking a look at the following expressions: $$ \sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} ...
6
votes
2answers
482 views

Indefinite integral question…

How can I solve this integral: $$\int \frac{1}{x\sqrt{x^2+x}}dx$$ I first completed the square and got: $$\int \frac{1}{x\sqrt{(x+\frac{1}{2})^2-\frac{1}{4}}}dx$$ Then I factored out 1/4 and got: ...
1
vote
1answer
45 views

Multi-index power series

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here ...
9
votes
1answer
252 views

Other integral related to Ahmed's integral

I have a doubt regarding the evaluation of the following integral : $$ \int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)} {(1 + 3x^2)\sqrt{1 + x^2}}\,du = ...
8
votes
1answer
129 views

How to integrate $\frac{x^{2}\log {\sin x}}{1+x^{6}}$

I recently stumbled upon a question $$\int_0^{\infty}\frac{x^{m-1}\log^{a}x}{1+x^n}dx$$ I was able to evaluate it,but I am curious if there exists a closed form for, ...
4
votes
1answer
87 views

How to integrate $\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t} dt$

This is a problem given in my homework . I have to find the integral$$\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t}dt$$ I am trying to use integral representation of the gamma ...
2
votes
1answer
37 views

Is there a closed form for a sequence invariant under “Cauchy square”?

For two sequences $a=(a_n), b=(b_n),$ define the Cauchy product as $a*b=(c_n),$ where $c_n=\sum_{k=0}^{n}a_kb_{n-k}.$ Then is there a closed form expression for a sequence $(a_n)$ whose product with ...
10
votes
0answers
481 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
2
votes
2answers
64 views

Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
6
votes
1answer
195 views

Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$

I would like to know if there exists a closed form for this integral $$\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx.$$ I tried the relation ...
3
votes
1answer
79 views

closed form for $\int_{0}^{1/2}\frac{x\cos\left(x\pi\right)^{2}\cos\left(2\pi kx\right)}{\sin\left(x\pi\right)}dx,k\in \mathbb{N}$

I would know if exists a closed form for $$\int_{0}^{1/2}\frac{x\cos^{2}\left(\pi x\right)\cos\left(2\pi kx\right)}{\sin\left(\pi x\right)}dx,k\in\mathbb{N}.$$I tried integration by parts without ...
5
votes
3answers
91 views

Closed form for the partial sum $\sum\limits_{k = 1}^n \frac{\ln k}k$

I'd like to find a closed form for this partial sum: $$\sum\limits_{k = 1}^n \frac{\ln k}k$$ Using the properties of the logarithms, I converted the above into $$\ln\left(\prod_{k = 1}^n ...