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 or simplification of a multiple definit integral of a product of a weight averaged parameters

I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} ...
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2answers
71 views

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
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0answers
11 views

Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
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1answer
23 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
5
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2answers
148 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
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1answer
40 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
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1answer
39 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
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5answers
74 views

Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
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1answer
94 views

Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
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0answers
30 views

Possible closed form or approximation?

Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this. $$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} ...
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36 views

Areas where closed form solutions are of particular interest

Assuming the definition of 'Closed Form' given in the table of: Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form? EDIT: Given the comment ...
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30 views

“Peak lemma” and explicit monotone subsequence

Looking at the proof of Bolzano–Weierstrass theorem, it found an interesting lemma (called the peak lemma here) : Every sequence $(x_n)_{n\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ has a monotone ...
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0answers
32 views

Different representation of $f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$

I am looking for a different way to calculate the following sum where $d,n\in \mathbb N$: $$f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$$ Here are some example results for different values of n ...
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94 views

Is there a closed form for these polynomials?

Let $P_0(x)=1, P_{-1}(x)=0$ and define via recursion $P_{n+1}(x)=xP_{n}(x)-P_{n-1}(x)$. The first few polynomials are $$ P_0(x)= 1\\ P_1(x) = x \\ P_2(x) = x^2-1 \\ P_3(x)= x^3 -2 x\\ P_4(x) = x^4 - ...
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1answer
31 views

Geodesic distance between equidistant points on a sphere [closed]

On the unit sphere equidistant points can be found for $1, 2, 3, 4, 6, 8, 12, 20$. The geodesic distance between the points are $\pi$ for $2$, $2\pi\over 3$ for $3$, $\pi\over 2$ for $6$, etc... Is ...
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1answer
21 views

When is a multiple sum given in closed form?

Let $d$ be a positive integer and $a>0$. Consider a following multiple sum: \begin{equation} {\mathcal S}^{(d)}_a(j) := \sum\limits_{0 \le j_1 \le j_2 \le \dots \le j_d \le j} \prod\limits_{l=1}^d ...
2
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1answer
36 views

Identifying closed form of two-parameter sequence

I've come across a two-parameter sequence $a_{nk}$ with $n=1,2,\ldots$ and $k=1,2,\ldots,n$, and I would like to identify a closed expression for it. So far I have the elements $a_{1k}=\{1\}\\ ...
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1answer
38 views

How to solve the recurrence relation $T(n)=aT(n-1)+bn^c$ with $T(1)=1$

How to solve this recurrence relation? $ T(n)=aT(n-1)+bn^c \\T(1)=1,$ where a, b, c are constant. I want to solve it using generating function, but get stuck. Could anybody help me?
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1answer
50 views

Solution of integral with exponential and trigonometric function

I have an integral, and it looks simple enough for me to believe it could have an analytic solution; however I am unable to find it. I was trying "Gradshteyn and Ryzhik's Table of Integrals, Series, ...
2
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0answers
24 views

Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when ...
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0answers
14 views

Getting closed-form for $f(x) = \sum_{k=0}^n (f(x-c_1 k-c_2)+1)$

I have to get closed-form for the recursive function $f(x) = \sum_{k=0}^n (f(x-c_1 k-c_2)+1)$ Where $c_1,c_2 \in \mathbb{N}$ $f(x) = 0 \,\,$ for $\,\, 0 < x < c_2$ $f(x) = -1\,\,$ for ...
2
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1answer
52 views

Vandermonde's identity and the close form of $\sum_{k=0}^r C(n,k) C(m,r-k) x^k$

I have a question related to Vandermonde's identity: From Vandermonde's identity, we have: $$ \binom{n+m}{r}=\sum_{k=0}^r \binom{n}{k}\binom{m}{r-k} $$ Now, I have an extra term $x^k$ inside the sum, ...
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2answers
56 views

Closed form of $\int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx, \end{equation} where $A$, $B$, ...
5
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1answer
136 views

hunting for the closed form of a series

Let $N$ be a positive integer, and $$ F(N) = \sum_{n=1}^{N} \frac{1-\cos\left(\frac{(2n-1)\pi}{2N}\right)} ...
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0answers
17 views

Is there a closed form of $n$ for :$h(n)=\frac{\sigma(n)}{n}$ for which $n$ is coprime to $\sigma(n)$?

It is well known that $h(n)=\frac{\sigma(n)}{n}$ is quit irrigulrar,I'm very interesting to know more about it's behavior and i would like to know more about coprimality characteristic then it must be ...
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1answer
65 views

Closed-form of a series relating to trigonometric function

Occasionally, we may meet some huge expressions, see $$ F(a)=\frac{1}{4}\sum_{n=1}^N \frac{\sin^2\big(\frac{(2n-1)\pi}{2N}\big)}{\Big[a^2-2a\cos \big(\frac{(2n-1)\pi}{2N}\big)+1\Big]^2} $$ where $a$ ...
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0answers
54 views

Potential function for an exact closed differential form and a solution of a corresponding linear first order homogeneous differential equation

A linear differential form $\sum_{i}\mathcal{E}_{i}(q)\, dq_{i}$ is an exact differential if the conditions $\partial\mathcal{E}_{i}(q)/\partial q_{j}=$ $\partial\mathcal{E}_{j}(q)/\partial q_{i}$ are ...
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1answer
54 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
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2answers
47 views

closed-form of an integral with regard to $a$ [closed]

Suppose that $a$ is a constant and $a>1$. So how can we evaluate the integral $$ I(a) = \int_0^1 \frac{t}{(a-t)\sqrt{1-t^2}}\;dt $$ I just wonder if there is a closed-form. Thank you.
2
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2answers
72 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
1
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1answer
74 views

I know the following integral can be computed in closed form, but I can't figure out how …

The following integral comes up for me when I'm computing a normalizing constant for a probability distribution: $$\int_0^\infty ...
14
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1answer
318 views

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

I was trying to find closed form generalizations of the following well known hyperbolic secant sum $$ \sum_{n=-\infty}^\infty\frac{1}{\cosh\pi ...
2
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2answers
60 views

Writing sigma notation $\sum^n_{i=1} \frac {i}{2^i}$ in closed form

What would be a way to find the closed form of $\frac {1}{2} + \frac {2}{4}+\frac {3}{8}+\cdots+\frac {n}{2^n}=\sum^n_{i=1} \frac {i}{2^i}=s$ I've looked at $\frac {s}{2}=\frac {1}{4} + \frac ...
5
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0answers
48 views

Non-existence of closed-form solutions

An equation like $$a^x+b^x=1$$ can be turned to the form $$t^\alpha+t=1$$ by a suitable change of variable. When $\alpha$ is a rational we can put that in a polynomial form $$u^p+u^q=1$$ and ...
4
votes
0answers
130 views

Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$

I seem to have seen quite a lot of integrals in the form: $$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$ But none of those hold a closed forms (at least to my knowledge) ...
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1answer
63 views

Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx, \end{equation} where $A$, $B$, and $C$ are ...
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2answers
170 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
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5answers
88 views

Quick way to get closed form for this recurrence?

Is there supposed to be a fast way to compute recurrences like these? $T(1) = 1$ $T(n) = 2T(n - 1) + n$ The solution is $T(n) = 2^{n+1} - n - 2$. I can solve it with: Generating functions. ...
13
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2answers
263 views

Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

I encountered the following integral and numerical approximations tentatively suggest that it might have a simple closed form: ...
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1answer
21 views

Closed form of this binomial expression?

Does a closed form for this binomial expression exists? $\sum_{K=2}^{N}\binom{N}{K}P^{K}(1-P)^{N-K}$ Thank you.
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2answers
100 views

Closed form of $\sum\frac{1}{k}$ where $k$ has only factors of $2,3$

Consider the set containing $A$ all positive integers with no prime factor larger than $3$, and define $B$ as $$ B= \sum_{k\in A} \frac{1}{k} $$ Thus, the first few terms of the sum are: ...
13
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4answers
329 views

Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$

I was able to calculate: $$\int_0^\infty\arctan\left(e^{-x}\right)\,dx=G$$ $$\int_0^\infty\arctan^2\left(e^{-x}\right)\,dx=\frac\pi2\,G-\frac78\zeta(3)$$ $G$ is the Catalan constant. In both cases ...
6
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0answers
226 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
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28 views

How to solve even/odd divide-and-conquer problems?

I am looking into something called the Josephus problem, which seems to be popular, so I am sure there are lots of explanations online, but I want to do the work myself, but I do need a small push to ...
1
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0answers
39 views

OEIS A249665 generating function

I'm stuck at finding the general term of the sequence $$1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \ldots$$ According to OEIS, Colin Barker conjectured the recurrence relation to be ...
0
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0answers
48 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
2
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1answer
115 views

Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
3
votes
0answers
108 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
5
votes
2answers
115 views

How to solve this integral $\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$

My classmate asked me about this integral:$$\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$$ but I have no idea how to do it. What's the closed form of it? I guess it may be related to the Airy ...
2
votes
2answers
108 views

Integrate the square root of the ratio of two quadratic polynomials

$$\int \sqrt{\frac{x^2+x-1}{x^2-1}} dx$$ I have been trying to find this integral for a while and I just can't. Does it even have a closed form?