Questions tagged [closed-form]

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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Closed-form for $F(n) = \sum_{d|n} \omega_x(d) $

Given $x\in \mathbb{C}$ let's define the function $$\cases { \omega_x(n)=1, \quad n=1 \\ \omega_x(n)=x^r, \quad n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}}$$ a) Prove that $\omega_x(n) \colon \mathbb{...
1
vote
1answer
34 views

Expressing the coefficients of $(1-x)^{1/4}$ using factorials

From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$ (1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n. $$ However, can I do the same ...
4
votes
2answers
167 views

Integral $\ 4\int_0^1\frac{\chi_2(x)\operatorname{Li}_2(x)}{x}\ dx+\int_0^1\frac{\log(1-x)\log^2(x)\log(1+x)}{x}\ dx$

A mind-blowing integral proposed by Cornel Ioan Valean. Prove that $$4\int_0^1\frac{\chi_2(x)\operatorname{Li}_2(x)}{x}\ dx+\int_0^1\frac{\log(1-x)\log^2(x)\log(1+x)}{x}\ dx=\frac{29}4\zeta(2)\...
1
vote
1answer
91 views

evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_n}{(2n+1)^3}$

where $H_n=1+\frac1{2}+\frac1{3}+...+\frac1{n}$ is the $n$th harmonic number. this sum was proposed by Cornel and I solved it using integration. can be solved using series manipulation? here is the ...
3
votes
0answers
97 views

Integrals involving powers and beta function

I have the three following integrals, very similar the one to the others, $$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$ $$I_2^{(p)}(...
5
votes
2answers
89 views

Solution of the integral $\int_1^\infty \frac{1}{x(e^{a x}-1)} dx$?

I am curious whether the integral $$\int_1^\infty \frac{1}{x(e^{a x}-1)} dx, \qquad a>0$$ has an exact solution. The best I can do so far is to see that is has the lower bound $$\int_1^\infty \...
2
votes
1answer
21 views

Sequences with reocurring/doubled elements: closed expression

I'd like to ask for your help in finding the closed-form expressions for the $n^{th}$ term of sequences with reoccuring or doubling items. Specifically the following sequences: $(a_n)=(0,1,1,3,3,6,6,...
12
votes
2answers
177 views

How to evaluate $\int_{0}^{1}\frac{\arctan x}{x} \log{\left(\frac{ 1+ x}{\sqrt{1+x^2}}\right)}\mathrm dx$

How to evaluate $$\int_{0}^{1}\frac{\arctan x}{x} \log{\left(\frac{1+ x}{\sqrt{1+x^2}}\right)}\mathrm dx$$ I tried to integrate by parts, but no way so far, help me, thanks.
3
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0answers
54 views

Extending differential form from a submanifold to a closed form

I am in $R^4$ in coordinates $x,y,z,t$. Can I extend an arbitrary 3-form defined only at points of the line $x=y=z=0$ to a closed 3-form in a neighbourhood of the line. In fact my question is only ...
5
votes
1answer
170 views

Summation closed form

I have the following sum, $$\sum_{j=0}^{\lfloor\frac{i+n-1}{n+2}\rfloor}(-1)^{j}\binom{n}{j}\binom{i+n-j(n+2)-1}{n-1}+\sum_{j=0}^{\lfloor\frac{i+n-2}{n+2}\rfloor}2(-1)^{j}\binom{n}{j}\binom{i+n-j(n+2)...
0
votes
2answers
52 views

$n$th derivative of $\sqrt{\frac{(s_0+T)(s_0 + T + 2\sqrt{a})}{(s+T)(s+ T+2\sqrt{a})}}$

I'm trying to obtain the $n$th derivative of $$f(s) =\sqrt{\frac{(s_0 + T + \sqrt{a})^2 - a}{(s + T+\sqrt{a})^2 - a}}=\sqrt{\frac{(s_0+T)(s_0 + T + 2\sqrt{a})}{(s+T)(s + T+2\sqrt{a})}}$$ that is, $...
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0answers
24 views

Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$

I am solving constrained polynomial systems resulting in constrained sums. I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in ...
2
votes
2answers
63 views

Bound on $x$ for $x-\ln x\ge 1+\epsilon$

I'm looking for a function $x(\cdot)$ where the domain is $\mathbb{R}^+$, that satisfies the following: For all $\epsilon > 0$, the inequality $x-\ln x\ge 1+\epsilon$ is satisfied for all $x \geq ...
0
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0answers
16 views

Closed Form Solution of Backward Heat Equation

The solution of the heat equation $$u_t = k u_{xx}, \; k>0$$ is given by convolving the initial heat distribution $u(x,0$) with the fundamental solution $$\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{...
6
votes
0answers
112 views

On the closed-form of the triple integral $\int_0^\infty\int_0^\infty\int_0^\infty\frac1{xyz\left(x+y+z+1/x+1/y+1/z\right)^2}\rm{dx\,dy\,dz}$

While doing research for my recent post on the Clausen function $\rm{Cl}_m(x)$, I came across in p. 19 of this paper (by one of the Borwein brothers) the remarkable integral, $$I_3 =\frac4{3!}\int_0^\...
2
votes
1answer
97 views

Double integrals involving incomplete beta function

I am trying to solve without success the following double integral $$I_1^{(p)}(N)\equiv\frac{1}{2^p}\int_0^1\text{d}x\int_0^1\text{d}y(1+y-x)^{N+p}(1+x-y)^{N-2}B\left(\frac{1}{1+y-x};N,p+1\right)\...
2
votes
1answer
95 views

A bug in Wolfram Alpha about an infinite series?

While verifying this MSE answer, I may have come across a bug in Wolfram Alpha. It evaluates the sum below as, $$\qquad A=\sum_{n=0}^\infty\frac{\binom{2n}{n}^2}{16^n(n+1)^3}=1.03928049\color{red}{51}...
2
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0answers
27 views

Closed form of $\eta^{(k)}(i)$.

Does anyone know closed form expressions for $$\eta^{(k)}(i)$$ up to high $k \in \mathbf{N}$? ($\eta$ is the Dedekind eta function.) For instance, I can use Mathematica to obtain $$\eta(i) = \frac{\...
0
votes
1answer
38 views

Closed form of series $z^n/n$.

Let $z\in\mathbb{C}$. In other question is answered precisely where $\sum\limits_{n=1}^{\infty} \frac{z^n}{n}$ converges. I have been looking for an expression of $$\sum\limits_{n=1}^{\infty} \frac{z^...
1
vote
1answer
99 views

Integral $\int_0^1 \sqrt{\frac{x^2-2}{x^2-1}}\, dx=\frac{\pi\sqrt{2\pi}}{\Gamma^2(1/4)}+\frac{\Gamma^2(1/4)}{4\sqrt{2\pi}}$

I have been trying to find the arc-length of $\sin^{-1}(x)$ over $[0,1]$. Of course, it is given by the integral $$J=\int_0^1\sqrt{1+\frac1{1-x^2}}\ dx=\int_0^1 \sqrt{\frac{2-x^2}{1-x^2}}\, dx$$ To ...
2
votes
0answers
50 views

How to analytically compute $\iiint \frac{1}{\sqrt{a+b \sin c}}\,da\,db\,dc$

I'm wondering whether the following integral can be analytically computed: $$\iiint\frac{1}{\sqrt{a+b \sin c}}\,da\,db\,dc$$ (Octave didn't yield any closed form associated with that.) Edit: $c$ ...
4
votes
2answers
99 views

Do I have a chance to get a closed form for this integral?

I conjecture that $$\int_{-z}^z \frac{\sin \left(\frac \pi {2z} x + \frac \pi 2 \right)} z \int_z^\infty \exp\left({- \frac {(y-x)^2 \pi^2}{16}}\right) \,d y d x \sim \frac 1 {z^2}$$ when $z\to \...
6
votes
3answers
152 views

Sum of reciprocal binomial coefficients

I am aware that $$\sum_{n=0}^\infty \binom{2n}{n}^{-1} = \frac{4}{3} + \frac{2\pi\sqrt{3}}{27}$$ though I do not know why it is true. More generally, I'm interested in the value of the series $$...
2
votes
1answer
44 views

Integrable subbundle

Let $D\subset TM$ is a integrable smooth regular subbundle and $f_{1},...f_{k}$ is smooth local frame for $Ann(D)$. Why $\omega=f_{1}\wedge ....\wedge f_{k}$ is closed form?
4
votes
1answer
117 views

Show that : $\displaystyle\int_0^{\infty}\frac{\ln(1+x^2)\operatorname{arc\,cot} x}{x}=\frac{π^3}{12}$

I need to prove this : $I=\displaystyle\int_0^{\infty}\frac{\ln(1+x^2)\operatorname{arc\mkern2mucot} x}{x}=\frac{π^3}{12}$ My try : $I\displaystyle\int_0^{1}\frac{\ln(1+x^2)\operatorname{arc\...
2
votes
2answers
69 views

Integral $E_n(a_1,…,a_n;t)=\int_{-\infty}^{\infty}\frac{x^2\cos(tx)}{\prod_{k=1}^{n}(x^2+a_k^2)}dx$

Evaluate $$E_n(a_1,...,a_n;t)=\int_{-\infty}^{\infty}\frac{x^2\cos(tx)}{\prod_{k=1}^{n}(x^2+a_k^2)}dx$$ for $i\ne j\iff a_i^2\ne a_j^2$, and $a_i\in\Bbb R\setminus \{0\}$. I've been able to ...
6
votes
0answers
89 views

Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?

(This summarizes my posts on Nielsen polylogs.) I. Question 1: How to complete the table below? Consider the special cases $z=-1$ and $z=\frac12$. Given the Nielsen generalized polylogarithm, $$S_{n,...
2
votes
3answers
114 views

Closed form for $\sum_{n=1}^\infty \frac{4^n}{n^p\binom{2n}{n}}$

By Mathematica, we find $$\sum_{n=1}^\infty \frac{4^n}{n^3\binom{2n}{n}}=\pi^2\log(2)-\frac{7}{2}\zeta(3).$$ How to find the closed form for general series: $$\sum_{n=1}^\infty \frac{4^n}{n^p\...
4
votes
1answer
118 views

Relating $\int_0^1\frac{(\ln x)^{n-1}(\ln(1-z\,x))^p}{x}dx$ and $\int_0^1\frac{(\ln x)^{n}(\ln(1-z\,x))^{p-1}}{1-z\,x}dx$

This post, after a complicated analysis, evaluates the integral $$I=\int_0^1\frac{\ln^2(x)\,\ln^3(1+x)}xdx$$ simply as $$I =-\frac{\pi^6}{252}-18\zeta(\bar{5},1)+3\zeta^2(3)\tag1$$ where, $$\...
14
votes
2answers
431 views

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

Prove $$\sf I=\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx=\frac{\pi}{6\sqrt{3}}\psi_1\left(\frac{1}{3}\right)-\frac{\pi^3}{9\sqrt{3}}-\frac{19}{18}\zeta(3).$$ I have thought about the integral ...
3
votes
1answer
96 views

Closed form for $\int_0^t(x+c)^p(1-2x)^{N-1}\text{d}x$

I am trying to find a closed form for the integral $$I\equiv\int_0^t(x+c)^p(1-2x)^{N-1}\text{d}x,$$ where $N\in\mathbb{N}$, $p>0$, $c\ge 0$ and $t\in\left(0,\frac{1}{2}\right)$. I thought to ...
2
votes
1answer
41 views

Finding closed form of exponential generating function involving identity permutation

Fix a prime number $p > 1$ and for a positive integer $n$, let $a_n$ be the number of permutations $π ∈ S_n$ such that $π^p = id$, where $id$ is the identity permutation. Find a closed form for the ...
6
votes
0answers
101 views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^5 2^n}$

Given the nth harmonic number $ H_n = \sum_{j=1}^{n} \frac{1}{j}$, we get from this post that apparently, $$\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n= S_{k-1,2}(z) + \rm{Li}_{\,k+1}(z)$$ for $-1\leq z\...
5
votes
1answer
111 views

On generalizing the harmonic sum $\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n = S_{k-1,2}(1)+\zeta(k+1)$ when $z=1$?

Given the nth harmonic number $ H_n = \sum_{j=1}^{n} \frac{1}{j}$. In this post it asks for the evaluation, $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\tfrac{5}{4}\zeta(4)$$ while this post and this ...
6
votes
5answers
223 views

Closed Form of $a_n = \int_0^1 \ln(1+x^n) dx$

I want to know the closed form of : $$a_n = \int_0^1 \ln(1+x^n)dx, \quad \forall n \in \mathbb{N}$$ I found : $$0<a_n<\frac{1}{n+1}, \quad \lim_{n\to\infty} a_n =0$$ I started from \begin{...
2
votes
2answers
110 views

Evaluate $\sum\limits_{n\geq1}\frac{(-1)^n}{3^n(2n+1)}\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}{n\choose k}(x^k-1)$

In my answer here, I reduce the problem of evaluating $$J=\int_0^{\pi/6}\frac{x\cos x}{1+2\cos x}dx$$ to the evaluation of $S(8-4\sqrt3)$, were $$S(q)=\sum_{n\geq1}\frac{(-1)^n}{3^n(2n+1)}\sum_{k=1}^...
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votes
1answer
15 views

Find the closed-form expression for an expression with finite and infinite sum

I'm trying to find the closed-form expression for the following equation: $$E[D] = \sum_{i=1}^{11}(11-i)(1-p)^{i-1}p + \sum_{i=12}^{\infty}(i-11)(1-p)^{i-1}p$$ My initial thought was to distribute ...
-1
votes
0answers
31 views

What's a compact way to express the product of terms in arithmetic progression?

There is a notation for the product of the first $n$ positive integers. My question is whether it is possible to express the product of any other non-trivial arithmetic progression with $n$ terms ...
0
votes
1answer
76 views

Recurrence $f_{n+2}=af_{n+1}+bf_n$

Solve the recurrence $$f_{n+2}=af_{n+1}+bf_n\qquad n\in\Bbb N_0\tag{1}$$ Where $a,b>0$ and $f_0,f_1$ are given. I know that if $$F_{n+1}=c_nF_n+d_n$$ then $$F_n=F_0\prod_{k=0}^{n-1}c_k+\sum_{m=0}...
2
votes
1answer
30 views

Taylor Type series

I'm stuck in the following series: $$\sum_{n=0}^{+\infty} \frac{1}{n!}\frac{d}{dx^n} \left( f(n-2x) \right) \left|_{x=0} \right.$$ where $f$ is a smooth function. At first glance it resembles a Taylor ...
2
votes
1answer
50 views

Exponential Type Series [duplicate]

I'm looking for a closed expression (if it exists) of the following sum: $$\sum_{m=0}^{\infty} \frac{m^n}{m!}c^m$$ where $n \geq 1$ is a positive integer, and $c$ is a fixed constant. The series seems ...
0
votes
2answers
44 views

Seeking a closed-form solution of a system of nonlinear trigonometric equations (converting Euler angles to spherical)

Following this and this questions, I want to solve the system of nonlinear trigonometrical equations (as part of an inverse kinematics calculation): $$\begin{align} \sin{\phi} \sin{\gamma} &= \...
0
votes
0answers
26 views

Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
0
votes
1answer
33 views

Closed form expression of a recursive relation

Consider the function $f \colon (0, +\infty) \rightarrow \mathbb{R} \colon x \mapsto 1/2(1 - e^{-1/x})$. Consider the recursive relation $$ \left \{ \begin{array}{ll} g_0(x) = f(x) & \\ g_k(x) = ...
5
votes
0answers
147 views

More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$

I. In this post, the OP asks about the particular log sine integral, $$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
5
votes
2answers
203 views

Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?

(This is related to this question.) Define the integral, $$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$ with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$, $$...
0
votes
1answer
81 views

Finding closed form for a product

I have no idea how to solve the below product whose closed form I need to solve a problem, can anyone at the very least guide me to a solution or give me a source to check? $\prod_{k=1}^{n}\dfrac{2^k-...
0
votes
0answers
23 views

The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...
0
votes
0answers
16 views

Radical Equation with Symmetry

Is there an approach to solve the following radical equation in $x$? $(P - 1)x = QR\left( Q + \sigma \sqrt{Q^2 + x} \right)\left( R + sign(P-1) \sigma \sqrt{R^2 + x} \right)$ We know $x\in (0,1)...
1
vote
2answers
28 views

Solving Recursion With Characteristic Polynomial

Say a linear recursion has the form: $a_n=a_{n-1}+a_{n-2}+2^{n-2}$, I know it would be possible to solve it with $a_n=a_{n-1}+a_{n-2}$ and $a_n=2^{n-2}$ and then combine the two using some coefficient ...