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
1answer
75 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
1
vote
1answer
14 views

Converting Recursive Function into Closed/Explicit Form

so I have this recursive function here: $\forall n>1,f(n) = 2(f(n-1)) + n-1$, (where it is $0$ when $n$ is less than $1$) So I have tried to use iteration for this but it just gets more ...
4
votes
0answers
61 views

Closed form for $\int_1^\infty\frac{dx}{\Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{dx}{\Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\Gamma(x)$ or the like) don't ...
4
votes
2answers
101 views
+100

Closed form of $I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
1
vote
0answers
23 views

Closed-form expression for this matrix equation?

I have the following matrices $P \in \mathbb{R}^{N \times N}$, $q(k) = \begin{bmatrix} q_1(k) \\ \vdots \\ q_N(k) \end{bmatrix}$. With $q_i(k) \in \mathbb{R}^n$ and thus $q(k) \in \mathbb{R}^{Nn}$. ...
0
votes
0answers
11 views

How to obtain closed form solution to the constrained optimization problem?

Suppose the following minimization problem: $$ N^*(\lambda)=\min_{X\in\mathbb{R}^8}\left\|D\left(A\cdot X-b\right)\right\|^2_2 \\ s.t. C_\lambda X= r_\lambda, $$ where $X\in \mathbb{R}^{8\times ...
4
votes
0answers
81 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
0
votes
1answer
29 views

Proving budget constraint is compact.

Given the prices $p \in \mathbb{R}_{+}^{k}$ and income $y \geq 0$, define the consumer's budget set as the set of feasible consumption bundles: $\beta(p,y) = \{x \in \mathbb{R}_{+}^{k}: ...
0
votes
1answer
28 views

Giving a closed expression to $\sum_{i=0}^b (-1)^{b-i} \binom{b}{i}\frac{1}{a+b-i}$

I want to prove $\sum_{i=0}^b (-1)^{b-i} \binom{b}{i}\frac{1}{a+b-i} = \frac{(a-1)! b!}{(a+b)!}$ yet I feel like I don't know how to even approach this problem. Any hints are welcome.
2
votes
0answers
26 views

Find spacing of fence posts with minimal distance to fence posts of two other fences

Definition 1: A "fence" is a set of "fence post positions", where each pair of adjacent positions has the same difference (the spacing), e.g. $\{1,2, 3, 4\}$. A fence is described by three values ...
3
votes
2answers
107 views

Proving the bound $\left ( 1+\frac{x}{n} \right)^n \leqslant 3^x$, $\forall x \in \mathbb{R^+}$

I'm trying to directly prove the above bound. I have tried expanding it $$\left ( 1+\frac{x}{n} \right)^n = \sum_{k\geqslant 0} \binom{n}{k}\left ( \frac{x}{n}\right)^k$$ $$= \sum_{k=0\dots n ...
6
votes
1answer
71 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
1
vote
0answers
14 views

Evaluation of infinite sum related to problems in elasticity

I'm working on some problems with relation to elasticity (plate mechanics in specific) and while I've made some progress the following sum is giving me a hard time ...
1
vote
1answer
25 views

What types of fractals have a closed-form interior formula?

I was looking at the Menger Sponge earlier, and I realized it has a neat property: Let x, y, and z be spatial dimensions, each between 0 and 1 (inclusive.) Express them as ternary floating point ...
11
votes
1answer
141 views
+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
1
vote
0answers
31 views

Explicit solution to a nonlinear equation possible here?

I am looking for a solution in $s$ to $$ \lambda -\frac{1}{s} +K e^t \log(\delta) \delta^s = 0 $$ Mathematica is not best pleased with this equation. If the equation were $$ 0- \frac{1}{s} +K e^t ...
7
votes
0answers
86 views

Known exact values of the $\operatorname{Li}_3$ function

We know some exact values of the trilogarithm $\operatorname{Li}_3$ function. Known real analytic values for $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
4
votes
2answers
53 views

Special values $\psi \left(\frac12\right)$ and $\psi \left(\frac13\right)$

I wonder if it is easy to prove that $$ \begin{align} \psi \left(\frac12\right) & = -\gamma - 2\ln 2, \\ \psi \left(\frac13\right) & = -\gamma + \frac\pi6\sqrt{3}- \frac32\ln 3, \end{align} ...
0
votes
1answer
25 views

Closed form for this incomplete gamma series?

The series I'm working with is $$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$ with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
0
votes
0answers
58 views

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

Looking for a closed-form of this integral. $$I=\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1+x^2}} dx$$ I'm looking for a closed-form of $I$ without using Meijer G-function, elliptic integrals or generalized ...
20
votes
2answers
376 views
+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
0
votes
0answers
26 views

Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
16
votes
3answers
254 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...
21
votes
3answers
388 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
6
votes
1answer
51 views

A closed form for $\sum_{k=0}^{n}(-1)^k\frac{k^p}{k!(n-k)!}$

Is there a closed form for $$ \sum_{k=1}^n (-1)^k\frac{k^p}{k!(n-k)!},\quad n=0,1,2\ldots,\,p=0,1,2\ldots. $$ I tried to identify the sum with Stirling numbers...
0
votes
2answers
55 views

Closed form for $1/(n (n + 1))$

What would the closed form be for $$ g(n) = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{n \cdot (n+1)}\;\;? $$ An example would be $g(3) = \frac{1}{2}+\frac{1}{6}+\frac{1}{12} = ...
2
votes
0answers
47 views

Closed form for the sum $\sum_{a=1}^{b} a^3\cdot (b \bmod a)$

How can we simplify $\sum_{a=1}^{b} a^3\cdot (b \mod a)$? For $a \ge \frac{b+1}{2} $ to $a = b$ it reduces to $$\sum_{a\ge \frac{b+1}{2}}^{b}a^3\cdot (b-a)=b\cdot\sum_{a\ge ...
2
votes
5answers
112 views

Closed form for $1 + 3 + 5 + \cdots +(2n-1)$ [duplicate]

What is the closed summation form for $1 + 3 + 5 + \cdots + (2n-1)$ ? I know that the closed form for $1 + 2 + 3+\cdots + n = n(n+1)/2$ and I tried plugging in $(2n-1)$ for $n$ in that expression, ...
12
votes
1answer
150 views

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

Using the method shown here, I have found the following closed form. $$ \int_0^{\!\Large \frac{\pi}{2}}\!\!\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan x}\right)^2\! \mathrm dx= ...
22
votes
4answers
606 views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
12
votes
3answers
350 views

Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$

$\qquad\qquad$ Is there any closed form expression for the imaginary part of $~\text{Li}_3\bigg(\dfrac{1+i}2\bigg)$ ? Motivation: We already know that ...
1
vote
1answer
27 views

simplifying an expression with even and odd integers

I got this expression for my $b_n$ to a Fourier series: $$b_n=\frac{(2- \pi^2 n^2)\cos(\pi n) -2}{4( \pi n)^3}$$ Now I want to write it in a closed form without the use of $\text{when } n \text{ is ...
6
votes
3answers
106 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
1
vote
1answer
52 views

Why is $\int_0^{2\pi} e^{i\,k\rho[\sin\alpha\cos\alpha-\sin\theta\cos(\phi-\beta)]}\mathrm{d}\beta = 2\pi J_0(k\rho\xi)$?

The following is an integral in Jackson Classical Electrodynamics (3rd ed.). In equation (10.112) the integral $$ \int_0^{2\pi} ...
3
votes
1answer
70 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
17
votes
2answers
282 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove\begin{equation} \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln(\cos x))^2 \right) \, dx=\pi\ln\ln2 \end{equation} I don't know how to answer it. When I asked this integral to my brother, ...
16
votes
1answer
211 views

Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$

This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here: Prove: ...
5
votes
1answer
80 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
4
votes
2answers
94 views

Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$

Is it possible to evaluate the following in a closed form? $$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$ I found the above definite integral at I&S but the solution is ...
6
votes
2answers
79 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
6
votes
4answers
167 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
4
votes
0answers
89 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
107 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
30 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 = ...
15
votes
2answers
99 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
101 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
141 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
57 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
21 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
28 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 ...