A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

learn more… | top users | synonyms

1
vote
1answer
46 views

How to find Real Part of PolyLog[3,(1-i)] in closed form

$ \Re \bigg(\text{Li}_3(1-i)\bigg)=\frac{\pi^a}{b}\ln(2)+\frac{c}{d}\zeta(e)$ has an approximate value of .8711588834109380 if $a=1 , b=-3415 , c=34 , d=39 , e=19$ are substituted into the closed ...
1
vote
2answers
42 views

Find another closed form for this sequence

I've recently come across a sequence while doing math which is: $$\{1;0;2;-1;3;-2;4;-3;...\}$$ And searching for a closed form to express the n-th therm I've came to the following: ...
2
votes
1answer
52 views

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ [duplicate]

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] ...
2
votes
2answers
42 views

Why is this the closed-form solution for this series? [duplicate]

I know this is simple, but I don't know very much at all about series, and I'm wondering how it's shown that: $$ 1 + 2 + 3 + \cdots + (n - 1) = \frac{n(n - 1)}{2} $$
2
votes
1answer
127 views

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} ...
4
votes
1answer
123 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{ ...
0
votes
0answers
25 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 ...
0
votes
1answer
59 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 ...
8
votes
2answers
228 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 ...
1
vote
1answer
46 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: $ ...
1
vote
1answer
48 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 ...
1
vote
5answers
102 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 ...
2
votes
1answer
110 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 $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
1
vote
0answers
35 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} ...
2
votes
2answers
43 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 ...
0
votes
0answers
31 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 ...
1
vote
0answers
38 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 ...
7
votes
3answers
119 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 - ...
1
vote
1answer
37 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 ...
0
votes
1answer
22 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
votes
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\}\\ ...
0
votes
1answer
41 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?
0
votes
1answer
54 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
votes
0answers
29 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 ...
0
votes
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
votes
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, ...
0
votes
2answers
58 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 $A$, $B$, ...
6
votes
1answer
142 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)} ...
1
vote
1answer
69 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 $a$ ...
1
vote
0answers
63 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 ...
1
vote
1answer
61 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" ...
-1
votes
2answers
48 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.
3
votes
2answers
77 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
vote
1answer
76 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 ...
15
votes
1answer
354 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
votes
2answers
70 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 ...
4
votes
0answers
55 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
141 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) ...
2
votes
1answer
71 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 ...
8
votes
2answers
179 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. ...
2
votes
4answers
106 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. ...
16
votes
2answers
346 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: ...
0
votes
1answer
24 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.
5
votes
2answers
103 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: ...
15
votes
4answers
361 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
votes
0answers
252 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 ...
1
vote
0answers
48 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
vote
0answers
46 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
votes
0answers
67 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
votes
1answer
135 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 ...