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".
3,350
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Evaluate $\sum_{n=1}^\infty \arctan\frac{1}{n^2}$
As shown here (Does $\sum_{n=1}^{\infty}\arctan(\frac{1}{n^2})$ converge?), the series $\sum_{n=1}^\infty \arctan\frac{1}{n^2}$ converges. It is similar to
$$\sum_{n=1}^\infty \arctan\frac{2}{n^2}=\...
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Why $\sum_{n=1}^\infty \arctan\frac{2}{n^2}=\frac{3\pi}{4}$ [duplicate]
How can I prove
$$\sum_{n=1}^\infty \arctan\frac{2}{n^2}=\frac{3\pi}{4}?$$
I'm supposed to use the identity
$$\arctan x+\arctan y=\arctan\frac{x+y}{1-xy}$$
where $xy\ne 1$ and $-\frac{\pi}{2}\lt \...
- 357
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votes
1
answer
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The limit of a Nasty Summation
I'm trying to evaluate the limit as h approachs 0 of the sum from k = 0 to n of:
$\frac{1}{h^n}(-1)^{k+n}\binom{n}{k}\frac{1}{(x+kh)^2-2(x+kh)+17}$
If it helps, it's the limit definition of the nth ...
- 65
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1
answer
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Continued fraction inequality with well know constant as $\pi$ and golden ratio.
I found this inequality beautiful let me share it :
Let :
$$d=\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\...
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How do I prove that an equation is bounded?
So I have a budget set that is B= {(x,y) |x ≥0,y ≥0,px(x) + py(y) ≤M}. How do i prove that is a closed and bounded set ?
I'm trying to maximise the utility function U (x,y) by choosing the optimal ...
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Does $\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdot\cdot\cdot\right)^{\frac{1}{x}}=L$ admits a closed form?
I try to simplify this limit :
$$\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdots\right)^{\frac{1}{x}}=L$$
Where we compose the Gamma function with itself .
From the past ...
- 3,413
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How to determine whether a given number is a member of given by the that recurrence sequence
Given recurrent sequence:
$x_1$;
$x_2 = (x_1^4 + 126x_1^2 - 1323)/8x_1^3$;
$x_3 = (x_2^4 + 126x_2^2 - 1323)/8x_2^3$;
...
$x_n = ( x_{n-1}^4 + 126x_{n-1}^2 - 1323)/8x_{n-1}^3$;
...
How to determine ...
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Closed form for an alternating values infinite sequence of nested roots
I tried to derive a closed form for an infinite sequence of nested square roots with alternating values and found myself with a 4th degree equation, which I'm fine with, but I was wondering if there's ...
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Is there a general form of a logical formula with N variables?
Let N = 2. Then there are 16 possible non-equivalent N variable logical formulas, listed below.
False,
A ∧ B,
¬(A → B),
A,
¬(B → A),
B,
A ⊕ B,
A v B,
¬(A v B),
¬(A ⊕ B),
¬B,
B → A,
¬A,
A → B,
¬(A ∧ B),...
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Closed form of the recursion formula for $\int_{0}^{1} \ln^r(1-x)x^n \, dx$
It is easy to see that we have the following recursive formula for the indefinite integral of log powers:
$$J_r := \int \ln^r(1-x) \, dx \stackrel{\text{sub.}+IBP}{=} -(1-x)\ln^r(1-x) - r\int \ln^{r-1}...
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Mean residual life closed form for burr distribution
I'm trying to find a closed form mean residual life of a Burr distribution with following survivor function:
$S(x) = (1+e^{(x-\mu) /\sigma})^{-k} $
where $\mu$, $\...
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1
answer
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Number of connected components for the filled julia set of $z^2 + c z^5$
For any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity
$$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\...
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answers
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Sum of $n$ degree derivatives taken w.r.t. different combinations of arguments
Consider $$ \Pi(b, s) = \int_{0}^{b} \chi(t)\ln(s-t) dt. $$
Notice that taking derivatives with respect to $s$ can help in obtaining formulae for integrals with the derivatives of $\chi$.
$$ \frac{d}{...
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answer
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Closed form of series $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$
I'm trying to get some closed form solution of $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$. To be precise gotten closed form should be analytic continuation for any given $x$. There ...
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Evaluate $\int_{0}^{\pi/2} \ln\left[ \tan\left ( \frac{\theta}{2}\right) \right ]^2 K\left ( \sin\theta \right )\text{d}\theta$
Let us define $K(x)$ as complete elliptic integral of the first kind, where $x$ is elliptic modulus. A possible closed-form is ($G$ denotes Catalan's constant.)
$$
\int_{0}^{\pi/2}
\ln\left[ \tan\left ...
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$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$ ??
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
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Is there a closed form of $\sum\limits_{r=0}^n \binom{n-r}{r}x^r$? [duplicate]
$\sum\limits_{r=0}^n \binom{n-r}{r}x^r=\sum\limits_{r=0}^{\lfloor n/2\rfloor} \binom{n-r}{r}x^r$
I need its closed form for a probability problem. I know about the case where $x=1$. It's the sum of ...
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2
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True or false: For continuous random variable $X$, if $E(X)$ has a closed form, then $P(X<E(X))$ has a closed form.
For this question, let us define "closed form" as an expression restricted to addition, subtraction, multiplication, and division; exponents and logarithms, including $e^x$ and $\ln{x}$; ...
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vote
2
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A certain sum of multinomial coefficients
I would like to know if there is a nice expression for the sum
$$
S(n)=\sum_{i+j=n}\binom{3i}{i,i,i}\binom{3j}{j,j,j}
$$
where $n$ is a non-negative integer. I have entered in the first few values of ...
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What's the average dimension of the Cantor sets?
Consider the open set $E=\{(x,y)\in(0,1)^2:x+y<1\}$ and define $\text{Cant}:E\to(0,1)$ such that $x^{\text{Cant}(x,y)}+y^{\text{Cant}(x,y)}=1$ which we may call the Cantor dimension function. My ...
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Is there a closed form of $\sum_{m,n=1}^\infty \frac{\cos(k m)}{(m^2+n^2)^{p/2}}$
I am looking for a closed form of
$$
\text{S}_p(k) = \sum_{m,n=1}^\infty \frac{\cos(mk)}{(m^2 + n^2)^{p/2}},
$$
where $p$ is a natural number. Some related functions are the Clausen functions
$$
\text{...
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Is there a closed form of the logarithmic infinite series $\sum_{n=0}^\infty x^n\log(1+x^n)$?
In a recent answer, I defined the function $\Psi(x)=\sum_{n=0}^\infty x^n\log(1+x^n)$ for $x\in(0,1)$ and proved the limit $\lim\limits_{x\to1^-}\left(\frac{1}{x}-1\right)\Psi(x)=2\log(2)-1$ by ...
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summation of $3^k$ [closed]
how do you write the closed form of a sum of the geometric progression of 3^n? Our teacher told us that $2^0+2^1.... 2^n$ is equal to $2^{n+1}-1$ but I am not sure how to apply that to a similar ...
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$\sum_{r=1}^\infty r^{-r}$? [duplicate]
is there any way to analytically find out at what value the summation of $r^{-r}$ from $r=1$ to infinity converges ?
it seems to be around 1.291 if i actually calculate it
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Do closed form expressions exist for $\int_0^1 \theta_3(x)\,dx$?
The Jacobi theta (or “thetanull”) function $\theta_3$ is defined by:
$$\theta_3(x)= \sum_{n \in \mathbb{Z}} \mathrm{e}^{-\pi n^2 x} = 1+ 2\sum_{n \in \mathbb{N}} \mathrm{e}^{-\pi n^2 x} \qquad \Re(x) &...
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Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$
Assume $c$ is a small real number.
QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$?
$$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$
I got as far as ...
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When is $\sin(\frac{\pi}{p}) $ not expressible by "real radicals" and $\sin(\frac{\pi}{q_i}) $? [duplicate]
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$
$\sin(\frac{\pi}{4})=\frac{1}{2}$
$\sin(\frac{\pi}{5})=\frac{\sqrt{5-\...
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votes
2
answers
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Imaginary part of dilogarithm
I have evaluated a certain real-valued, finite integral with no general elementary solution, but which I have been able to prove equals the imaginary part of some dilogarithms and can write in the ...
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Finding the closed form of a polynomial defined recursively
This is my first ever question on this platform
I want to find the closed form for the polynomial $J_j(x)$.
This polynomial is defined using this recurrence relation:
$$ J_j(x) = \frac{x^{2j+2}}{2j+2}...
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Reference for, and/or proof of, $\prod_{n=1}^\infty(\frac{4n+1}{4n-1})^{4n}(\frac{2n^2-2n+1}{2n^2+2n+1})^n=\sqrt2\cosh(\pi/2)e^{-2G/\pi}$
Context:
I have derived some infinite products that I think are not well known. This is the easiest of them:
$$\prod_{n=1}^{\infty}\left(\frac{4n+1}{4n-1} \right)^{4n}\left(\frac{2n^2-2n+1}{2n^2+2n+1} ...
user1142168
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Expressions for $\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}$
It is known that $$\delta=-e\left(\gamma+\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}\right)$$Where $\delta$ is the Euler-Gompertz constant and $\gamma$ is the Euler-Mascheroni constant. I want to find ...
- 3,394
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votes
2
answers
82
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Number of Edges in a Subset Graph of a Power Set
Given a set $A$ of cardinality $n$, let $\mathbb{P}(A)$ be the power set of $A$. What is the number of edges of the intersection graph of the powerset of $A$. I.e how many pairs of sets $x,y$ have the ...
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Sums with nested radicals related to $1/\pi^2$
Context:
I evaluated some sums related to $1/\pi^{2}$ that for a reason I don't know I can't find in the literature.
$$(1)\hspace{.5cm}{1\over 2\sqrt{2}}+{1\over 2^{3}\sqrt{2+\sqrt{2}}}+{1\over 2^{5}\...
user1142168
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Why certein form of solution of heat equation cannot be used to estimate flux
Considering the 1-D semi-infinite domain,
$$
\begin{aligned}
& \frac{\partial^2 T(x, t)}{\partial x^2}=\frac{1}{\alpha} \frac{\partial T(x, t)}{\partial t} \quad \text { in } \quad 0<x<\...
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Turn a real number (with a complex closed form) into its trigonometric form
This post outlines 'fake' complex numbers (real numbers with complex closed form that usually come from the roots of unfactorable cubics (the example I need right now), or they can come from things ...
user1112591
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Prove that $\int_{0}^{1}x^2K^\prime(x)^3\text{d}x =\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2} -\frac{\pi^4}{40}$
I need to prove the following result
$$
\int_{0}^{1}x^2K^\prime(x)^3\text{d}x
=\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2}
-\frac{\pi^4}{40},
$$
where $K^\prime(x)=K\left(\sqrt{1-x^2}\right)$...
- 3,299
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votes
2
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119
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Semantic meaning of 'finiteness' in closed forms [closed]
Definition by Wolfram MathWorld: "An equation is said to be a Closed-form Solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted ...
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votes
2
answers
48
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Closed Form of the Chebyshev Polynomials of the First Kind [Proof Request]
$$T_n(x) = \frac{n}{2} \sum_{r=0}^{\lfloor \frac{n}{2} \rfloor} \frac{(-1)^r}{n-r} \binom{n-r}{r} (2x)^{n-2r}$$
Searching on the web yielded no results, and the result is given without proof on OEIS ...
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1
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Other closed forms of $\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)$
It is known that
$$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\beta$$converges. I wonder if there are any other closed forms for this limit. At first I ...
- 3,394
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0
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Interpretation of regular expression into closed form
I have this example of regular expression :
$*α* = (hh)* ((b|b)d)* $and want to convert it into
so is it wrong when I convert it into
{$ {(h)ⁿ + (b)ⁿ |d^m ≥ 0,n ≥ 1 } $} or I missed somthing ?
...
- 1
3
votes
2
answers
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What is the rigorous definition of a closed-form function?
I have been wanting to ask this question for a long time. In many fields of mathematics, mathematicians are interested in whether some function $f$ is a closed-form function. However, I have never ...
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Have you ever been very surprised that something has, or doesn't have, a closed form? [closed]
I'm trying to develop my intuition about when something likely has, or does not have, a closed form expression. So I would like to ask:
Have you ever been very surprised that something has, or doesn'...
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vote
2
answers
156
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Closed form for solution of $\ln(x)\ln(x+1)=1 $ [closed]
Is it possible to find a close form for the solution of this equation (maybe with the use of Lambert W function)?
$$ \ln(x)\ln(x+1)=1 $$
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votes
1
answer
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Closed form of a geometric series without some term
I have to study find the closing form of the following series:
\begin{equation}
f(1) + f(2) + f(3) + f(5) + f(6) + f(7) + \dots
\end{equation}
So basically the sum of all terms without the multiple of ...
0
votes
0
answers
41
views
Inner product of 4 Legendre Polynomials
Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner ...
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1
vote
0
answers
28
views
Explicit solution to a singular linear system
A quadratic optimization problem I have leads to a system of linear equations which I want to solve explicitly (with a closed formula). We are searching for $(s_{1},s_{2},s_{3},...,s_{N})$ such that
$$...
- 1,413
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votes
1
answer
81
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Is there any form of closed form solution for these two integrals?
I have two integrals of the from
\begin{equation}
\int_{-\pi/2}^{\pi/2} \mathrm d k e^{-i\omega \sin k} \frac{\lambda}{\alpha+\beta \cos 2k}
\end{equation}
and
\begin{equation}
\int_{-\pi/2}^{\pi/2} \...
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0
answers
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Closed form of $\int_0^{\pi\text{ or }\frac\pi2}\cos(w (\cos(t)+a t-b))dt$.
Although an integral for $x=\dots$ exists, it is slightly harder to integrate. Dirac $\delta(t)$ helps solve $\cos(x)+ax=b$: $$\frac1{\sin(x)-a}=\int_a^b \delta(\cos(t)+at-b)dt\tag1$$ From numerical ...
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What is the determinant of the matrix of $A_{ij} = \gcd(i,j)$ for $i,j$ ranging from $0$ to $n−1$?
The determinant of the matrix whose entries are $\gcd(i,j)$ for $1≤i,j≤n$ equals $\prod_{k=1}^n \varphi(k)$ where $\varphi$ is Euler's totient function: see A001088 in the OEIS, as well as the paper “...
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0
answers
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Computing $\frac{d^{n-1}}{dz^{n-1}}\frac{z}{\ln\left(z!\right)}^{n}$
I am trying to compute
$$\frac{d^{n-1}}{dz^{n-1}}\left(\frac{z}{\ln\left(z!\right)}\right)^{n}$$
The problem arises when dealing with inversion formulae. My question is, can this expression be ...
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