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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22
votes
4answers
296 views

How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

How to compute the integral, $$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$ where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?
4
votes
0answers
123 views

Is there a closed-form expression for Shapley value of glove game?

Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number ...
1
vote
1answer
29 views

Solution to a pde

I have a PDE system that I am trying to solve at steady state. When I make the appropriate substitutions, I get an equation of the following form: $$\frac{1+M}{M}\frac{d^2 M}{d x^2}=1$$ Is there a ...
2
votes
2answers
37 views

Help with recurrence $T(n) = T(n/2) + n$

I just need help seeing where I went wrong in this solution. $$T(n) = T\left(\frac{n}{2}\right) + n,~~~ T(1) = 0$$ By master theorem, this is $\theta(n)$. However, when I try to solve it, it ...
24
votes
4answers
478 views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
1
vote
0answers
21 views

This one weird infite product can define exponentials in terms of itself. What does it do for other constants?

What is... $$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ This is similar to my other question. However, there is a constant factor rather than variable in the ...
9
votes
1answer
183 views

Closed form for ${\large\int}_0^\pi\frac{x\,\cos\frac x3}{\sqrt[3]{\sin x}}dx$

I'm trying to find a closed form for the integral below and I found the following conjecture using computer search (and some lucky guesses): $$\int_0^\pi\frac{x\,\cos\frac x3}{\sqrt[3]{\sin ...
4
votes
2answers
194 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
0
votes
2answers
49 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
2
votes
3answers
98 views

Proving $\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt \to \gamma$

I have to prove that $\displaystyle \lim_{n\to\infty}\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt= \gamma$ I tried to expand $\left(1-\frac{t}{n}\right)^n$ and swap sum and integral, which ...
0
votes
1answer
51 views

Closed Form Solution for Minimization involving Standard Normal CDF and PDF

Could someone please advice and provide detailed steps regarding any possible closed form solutions or other suggestions regarding solving a minimization problem of the type shown below? Here, $\phi$ ...
0
votes
1answer
50 views

solve logarithmic equation without numerical methods

Is there algebraic method to solve following equation for $x$: $$ a x + b \ln x + c = 0 $$ with $a , b , c$ constants without using numerical methods and ln means natural logarithm.
14
votes
1answer
207 views
+50

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
5
votes
1answer
134 views

How to use Fourier Transform with non-trivial boundary conditions such as in potential flow around a plate?

I'd specifically like to be able to solve this PDE with boundary conditions corresponding to flow around a line (plate cross-section), otherwise known as flow-tangency, with integral transforms. ...
1
vote
2answers
131 views

Closed form of partial hypergeometric sum

Can we get closed form for $$\sum_{k=0}^m \left(-\frac12\right)^k \binom{2m}{m-k}k^p,\quad p\in\mathbb{N}\,?$$ In Concrete Mathematics Knuth describes Gosper's algorithm and its Zeilberger's ...
19
votes
2answers
317 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,dx$ have a closed form?

Mathematica gives an approximate result of $1.581949621806183890451628...$, but no exact form. I predict it's a function of $e$ and $\pi$, and perhaps even the Golden Ratio $\phi$ (It certainly ...
0
votes
0answers
27 views

Series identity for cotangent

How to prove that $x \cot(x) = 1 - 2 \sum_{n=0}^{\infty}{\frac{x^{2}}{(n \pi)^{2}-x^{2}}}$? First, it does not seem to be solvable, using considerations regarding Taylor series. The Fourier approach ...
0
votes
1answer
87 views

About a sum involving factorials.

I would like to know if there is a closed form of $$\sum_{k=0}^{n}\frac{4^{k}}{\left(2k\right)!\left(n-k\right)!^{2}}.$$ Wolfram gives a strange closed form and, i.e., ...
13
votes
2answers
558 views

Infinite sum of reciprocals of pentagonal numbers

How do I find this sum: $$\sum_{n=1}^\infty \frac{1}{p(n)}$$ where $p(n)=\dfrac{n(3n-1)}{2}$ is the $n$th pentagonal number? I know it is a convergent series, but I don't know if the sum can be ...
-2
votes
1answer
53 views

Website with possible closed forms of numbers

I encountered a website that had a large number of possible closed forms per a user number entry. It is not WA. I cannot locate it now. I had it saved before having to reinstall my browser. Anyone ...
6
votes
0answers
108 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
5
votes
2answers
149 views

Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

In this question, Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$ , robjohn evaluates the integral to a nice summation with an approximate value. When plugged into W|A, it ...
2
votes
0answers
33 views

Closed form of a matrix product

Is there any closed form or a bound for a matrix product of this kind $$ P=\prod_{i=1}^n \begin{pmatrix} 1-a & a \\ b_i & 1-b_i \end{pmatrix}, \quad a,b_i \in [0,1] $$ for an arbitrary ...
1
vote
2answers
33 views

Is there a closed-form solution to the following equation?

I would like to know if there is a closed-form solution for $x$ in the following equation. If there is no such form, how can you show this? ...
0
votes
0answers
13 views

Fourier Transform of $x^p \cdot {{df^q} \over {dx^q}}$

What is the Fourier Transform of $x^p \cdot {{df^q} \over {dx^q}}$? This seems like an elementary question, but my CRC book of standard formulae doesn't have it. My attempt is rather trivial, but for ...
20
votes
2answers
475 views
16
votes
0answers
191 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
2
votes
1answer
53 views

Explicit solution of parametric solutions of an ODE

I need to find the explicit solution of the following ODE: $y'+\sin y'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+\sin t$ and $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...
6
votes
0answers
60 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
13
votes
3answers
268 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
2
votes
3answers
47 views

Determine a closed form for this sequence

Every year, 38 % of the amount of fish in a pond die. The 1st of May 2011 there were 5200 fish in the pond. Every year after May 1st 2011, 1900 new fish are added to the pond. Let $a_n$ be the amount ...
9
votes
3answers
227 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
1
vote
1answer
45 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
4
votes
2answers
100 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
5
votes
0answers
144 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
2
votes
0answers
61 views

Closed form for sequence: $\sum_{j=1}^k 2^j j^{1/2}$

Any idea how to find the closed form for either of following sequences: $$ A(k) = \sum_{j=1}^k 2^j j^{1/2} $$ or $$ B(k) = \sum_{j=1}^k 2^a, \quad \quad a:= {j^{1/2}} $$ Note: closed form for one of ...
3
votes
1answer
52 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
0
votes
2answers
33 views

Solving an inequality involving a floor

Increasing the integer $k$, I can make the floor of $L/k$ smaller than $r$: $$\left\lfloor \frac{L}{k} \right\rfloor \lt r$$ where $L, k, r$ are positive integers, $k\leq \lfloor \frac{L}{2} ...
13
votes
2answers
225 views

Closed form $\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx$

Does the following integral $$\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx, \; \; n \in \mathbb{N}$$ have a nice closed form? Basically I cannot tackle it in any ...
1
vote
3answers
89 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
11
votes
3answers
139 views

Finding this summation: $\sum_{n=1}^{\infty}\frac{(2n+99)!(3n-2)!}{(2n)!(3n+99)!}$

What would be an easy method to find the approximate value of/close the form (the former will work too, if reasonably correct, a few decimal places, not more than that.): ...
6
votes
1answer
123 views

A difficult integral: $\int_0^{+\infty} e^{ - x}\left(\frac{1}{x( e^{ - x} - 1 )} + \frac1{x^2} + \frac1{2x} \right) \, dx$.

Could you help me calculate the integral? $$\int_0^{+\infty} e^{ - x} \left( \frac{1}{x( e^{ - x} - 1)} + \frac{1}{x^2} + \frac{1}{2x} \right) \, dx .$$
1
vote
2answers
103 views

Closed expression of the following integral?

I believe that the following integral has a closed expression, but I haven't been able to check it $$I(k)=\int_{-\infty}^{\infty}dt\,\text{erf}\left(\frac{t}{b}-i \frac{1}{2}b(k+a)\right) ...
0
votes
0answers
21 views

Closed form for sum resembling generating function

Is there a general closed-form solution for $$\sum_{k=0}^\infty \frac{f(k)}{z-k}$$ as a generating function of $f$? It vaguely reminds of a couple of other kinds of generating functions, but not in ...
2
votes
2answers
70 views

How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...
4
votes
0answers
37 views

How do I find the finite limits of this infinite product?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ I'd like closed form solutions, and in this case that means any ...
2
votes
1answer
63 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
0
votes
1answer
46 views

Probability that a random graph is connected

Let $V=\{v_1,\dots,v_n\}$ a set of $n$ vertices. Define $\mathcal{G}$ to be the set of all graphs on $V$. $|\mathcal{G}|=2^{\binom{n}{2}}$. What is the probability that a random graph from ...
1
vote
2answers
54 views

Calculate the sum of $\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n}$

$$\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n} = \sum_{n=1}^{+\infty} \frac{1}{3^n} + \sum_{n=1}^{+\infty} \frac{2^n}{3^n}$$ Each term is geometric series with $-1<r<1$ so they are all covergent. As ...
0
votes
3answers
31 views

Linear recursive sequence in closed-form function

I've been trying to find an answer for a question for some time, and I've done some Google searching but can't seem to figure out exactly how to solve it. It is a linear recursive sequence, and it ...