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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-1
votes
1answer
46 views

Integral involving the error function

Is there a closed form solution to the integrals \begin{align} I_{c} &= \int_{0}^{\infty} \cos(a x) \, \operatorname{erf}(b x) \, dx \\ I_{s} &= \int_{0}^{\infty} \sin(a x) \, ...
12
votes
3answers
285 views

Integral $\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}}\operatorname d \!x$

Could you please help me with this integral? $$\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}} \operatorname d \!x$$ Update: user153012 posted a result given by a computer that contains scary Appel ...
0
votes
1answer
41 views

How to write in closed form this nasty expression?

I have something like $$ v_1 l_1+v_1 l_2+ v_2l_1+v_2l_2$$ and I am trying to write it in closed form as such, $$\sum_{j=1}\sum_{i=1}v_il_j$$ I know this is not right but I want something like that. ...
6
votes
1answer
206 views

Prove ${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits. ...
1
vote
0answers
31 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} ...
8
votes
1answer
568 views

A long nasty limit problem

Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$$ My professor gives this ...
0
votes
1answer
45 views

How to find this integral $I=\int_{0}^{1} t^{-1/2}\left(1-\frac{t}{1+4y}\right)^{-1/2-ix}dt$

Question: Find the closed form of $$I(x,y)=\int_{0}^{1}t^{-1/2}\left(1-\dfrac{t}{1+4y}\right)^{-1/2-ix}dt$$ where $i^2=-1$ I have used Wolfram Alpha but it can't help me out. How to ...
2
votes
3answers
49 views

How to derive the closed form of this recurrence?

For the recurrence, $T(n) = 3T(n-1)-2$, where $T(0)= 5$, I found the closed form to be $4\cdot 3^n +1$(with help of Wolfram Alpha). Now I am trying to figure it out for myself. So far, I have worked ...
20
votes
1answer
456 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
8
votes
1answer
188 views

Generalized FoxTrot Series $F(a,b,q,x) = \sum_{k=q}^{\infty} \dfrac {(-1)^{k+1} k^a}{k^b+x}$

The FoxTrot Series is defined as: $$F = \sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1}.$$ Using partial fraction decomposition we can show that $$F = \frac 13 \left[ 1 - \ln2 + ...
2
votes
2answers
40 views

Sums with squares of binomial coefficients multiplied by a polynomial

It has long been known that \begin{align} \sum_{n=0}^{m} \binom{m}{n}^{2} = \binom{2m}{m}. \end{align} What is being asked here are the closed forms for the binomial series \begin{align} S_{1} &= ...
8
votes
1answer
71 views

A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $

This is another 'arctanlog' series: $$ S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $$ Maybe differentiating with respect to some parameter could be of interest. What ...
13
votes
1answer
299 views

how to calculuate $\int_0^ \pi \sqrt{1+x^2 \sin^2x}dx$

I was finding arc length of $y=\sin x - x \cos x$ $(0 \leq x\leq \pi)$ and I found I've to solve $$\int_0^\pi \sqrt{1 + x^2\sin^2{x}}\, dx $$ but I have no idea about this. I tried using $\sin^2x ...
3
votes
2answers
83 views

Closed form for $\sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}}$

Can anybody find a closed form for this infinite sum? $$S = \sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}},$$ where $\zeta$ is the Riemann ...
0
votes
1answer
39 views

Solving recurrence relation in form of $f(n)=f(n-1)+K-n$

I asked a question (now deleted it) on finding a relation between $$4,10,15,\cdots$$ I studied a a little about recurrence relation and solving them. for the above sequence I observed that $$f(n) = ...
2
votes
1answer
65 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
8
votes
1answer
137 views
+50

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
5
votes
4answers
247 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
1
vote
3answers
49 views

Yet another sum involving binomial coefficients.

Given $A,B,N \in \mathbb N$ Is there a closed form for this expression? $$\sum_{n=1}^N n \binom{A}n \binom{B}{N-n} $$ If there is such, can you give a proof? EDIT: $A,B \geq N$
4
votes
1answer
81 views

How to evaluate the following infinite sum involving the Riemann zeta function?

I want to find a closed form for the following infinite sum: $$\sum_{k=2}^{\infty} \frac{(-1)^k\cdot(k-1)}{k\cdot(k+1)}\cdot \zeta(k)$$ Is it possible? My approach was to transform it into a double ...
0
votes
1answer
17 views

Antisymmetric asymptotic curve with only simple binary arithmetic?

I'm looking for an s-curve formula with similar properties to $Sigmoid$ or $\tan^{-1}$, but without 'expensive' unary functions or their binary generalizations (e.g. $^x\log y$). The only allowed ...
5
votes
1answer
83 views

Integrals $\int \frac{1}{\operatorname{arctanh}(x)} \, dx$ and $\int \frac{1}{\operatorname{arccoth}(x)} \, dx$

Do we know anything about this integrals? $$ \begin{align} I_1(x) = \int \frac{1}{\operatorname{artanh}(x)} \, dx \\ I_2(x) = \int \frac{1}{\operatorname{arcoth}(x)} \, dx \end{align}$$ Similar ...
1
vote
2answers
79 views

Solving equation involving binomial function

Solve for $x$ in terms of $i$ and $j$: $$ \binom{x}{i} = j $$ where $x$ is Real; $i$ and $j$ are Integers: $x \geqslant i$, $i \geqslant1$, $j \geqslant 0$. I came across this problem while trying ...
1
vote
1answer
41 views

How to calculate convolution of two logarithmic function, i.e. $\int \ln(\tau)\ln{(x-\tau)}d\tau$

Here I have a problem to calculate the probability density function (PDF) of the sum of two independent random variables (RVs), $Z_0,Z_1$, and the PDF of $Z_0,Z_1$ are as follow: ...
15
votes
2answers
211 views

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

One of the possible ways of computing the series is to obtain the generating function, but this might be a tedious, hard work, pretty hard to obtain. What would you propose then? ...
1
vote
1answer
41 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
5
votes
0answers
126 views

Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
1
vote
2answers
53 views

How do you solve these recurrence relations for a closed form?

I'm not sure what methods are used to solve recurrence relations for a big-$O$ notation. Thinking about the problem conceptually doesn't really help me, but I feel like I could use some form of ...
4
votes
1answer
75 views

General term of $(1+x)(1+x^2)(1+x^3)…$?

Is there a closed for the coefficient of $x^n$ in $(1+x)(1+x^2)(1+x^3)\cdots$? If not, then what is the closest to a closed form that anyone has found? (An infinite series that approximates it ...
2
votes
1answer
171 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
2answers
41 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 ...
6
votes
1answer
177 views

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

Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of ...
9
votes
2answers
258 views

Closed form of $\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 ...
0
votes
0answers
12 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 ...
6
votes
1answer
232 views

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
55 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
29 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.
3
votes
1answer
47 views

Spacing of fence posts with minimal distance to other fence posts

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
109 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 ...
18
votes
2answers
335 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where ...
2
votes
1answer
33 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
26 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 ...
16
votes
1answer
308 views

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 ...
8
votes
0answers
116 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 $\operatorname{Li}_3$: $\operatorname{Li}_3(-1)=-\frac{3}{4} \zeta(3)$ ...
6
votes
2answers
74 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} ...
1
vote
1answer
33 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 ...
21
votes
4answers
662 views

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
32 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 ...
18
votes
3answers
279 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 ...