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".
3,693
questions
0
votes
0
answers
53
views
$ \lambda^{*}(n) $ minimal polynomials
I already asked a closely related question on MSE.
Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.
Is there a way to calculate the ...
- 923
2
votes
0
answers
238
views
Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$
Context
$\begin{align}
K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}
\end{align}$
and
$\begin{align}
E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2}
\end{align}$
the complete elliptic ...
- 323
9
votes
1
answer
384
views
evaluation of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n} H_{n+1}^{(2)}}{(n+1)^{2}}$ and other Euler sums
I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$
Here is my attempt so solve the integral
\begin{align}
&\int_{0}^{1} \frac{\ln (x) \ln^{...
user1313975
7
votes
1
answer
295
views
how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$
Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$
MY try to evaluate the integral
$$
\begin{aligned}
& I=\int_0^{\infty} \frac{x \...
user1313975
7
votes
3
answers
228
views
Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?
Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ?
Context
Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
- 25.7k
2
votes
0
answers
87
views
Can multiary compositions of elementary functions have an elementary inverse?
I'm looking for general methods for solving equations of elementary functions of one variable in closed form.
Definition:
The elementary functions are generated by applying finite numbers of $\text{...
- 7,212
1
vote
0
answers
51
views
Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$
I am now trying a direct approach to solving my question about
$$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$
where the $a_i$ are all positive. Note that the $\arctan$s ...
- 104k
3
votes
0
answers
53
views
Find $\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$ algorithmically
It sometimes happens that
$$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$
is algebraic for positive integers $m,n,a_k,b_k$. For example,
$$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
- 667
2
votes
1
answer
78
views
How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t $
I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm
d}x .$
On the path of integrating the main function, I am stuck at this integral. I don't know how ...
4
votes
3
answers
220
views
Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$
Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$
Some thinking before trying
At least we ...
0
votes
0
answers
22
views
Is there a closed form for $\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$? [duplicate]
$$B(a,b):=\int _0 ^1 t^{a-1}(1-t)^{b-1}dt$$
What happens If we change the negative sign to positive ?
$$F(a,b):=\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$$
This question came to me while solving this limit
$$...
- 6,620
8
votes
1
answer
285
views
Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$
with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$
$$I(a,b)=
\frac\pi4\left(\frac{\pi^2}6
-\Li\...
- 104k
0
votes
3
answers
75
views
How do I calculate the closed form of $\sum_{k=2}^\infty kx^{k-2}$
This is an exercise from Wade, the answer is given as; $$\sum_{k=2}^\infty kx^{k-2}=\frac{2-x}{(1-x)^2},$$ but there is no help as to how to arrive at that answer. I have completed the first question ...
- 3
3
votes
1
answer
128
views
Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)
I'm looking for ways to compute the coefficients of the power series
$$
\sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k
$$
(a prior version of the question asked whether such an ...
- 40.1k
0
votes
0
answers
22
views
Solve Minimax Rules in Finite Case
Let $\Theta = \{\theta_1, \cdots, \theta_n\}$ be the space of parameters and $D = \{d_1, \cdots, d_m\}$ be the space of decisions (that is, they are arbitrary finite sets with at least two elements). ...
- 2,497
3
votes
2
answers
89
views
Integral in terms of Hypergeometric function
Consider the integral $$ I = \int_0^1 \int_0^1 (1+c^2v^2)^{-s}u^{1-2s}(1-uv)^{s-1}dudv$$
where $c>0$ is some constant and $0<s<1$. Clearly the integral is absolutely integrable (Two ...
- 3,360
0
votes
0
answers
74
views
Is it possible to find a closed form for $i!$? [duplicate]
I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any.
$$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$
$$i! =\lim_{n \to \...
12
votes
1
answer
653
views
Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.
There is numerical evidence that
$$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$
How can this be proved?
Context
In another question, three random ...
- 25.7k
6
votes
0
answers
94
views
Is there a closed-form expression for this iterated mean?
Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers:
...
- 15.8k
5
votes
2
answers
105
views
Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
- 1,258
6
votes
2
answers
261
views
Find the closed form of $_3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)$
Context
Some investigation suggests that the following identity is true:
\begin{align}
_3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)=\frac{3\sqrt{2}\sqrt{\pi}\left(2\log({1+\...
- 323
4
votes
2
answers
238
views
Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $
Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
- 6,620
20
votes
1
answer
1k
views
Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.
There is numerical evidence that
$$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
How can this be proved?
...
- 25.7k
6
votes
2
answers
141
views
How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum ...
- 6,620
12
votes
2
answers
504
views
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
- 3,881
3
votes
1
answer
196
views
Property of $a_{n+1} = a_n - \frac{1}{a_n}$
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 = k >0$. Prove that if the first $n$ such that $a_n \leq 0$, then $n \in O(k^2)$.
I ran a computer simulation, and it seems ...
- 85
0
votes
0
answers
83
views
Finding closed-form maximum of a function with no obvious root
I have the below function:
$$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
1
vote
1
answer
123
views
Is there a way to solve quintics of the form $ax^5+bx^4+c=0$?
It is known that if we only consider the first four decimal digits of the following (no rounding), we have that this holds:$$\pi^4+\pi^5=e^6\label1\tag1$$However, if we use even one more decimal place,...
- 2,652
7
votes
1
answer
155
views
Is there a closed-form expression for the series $\sum_{n=0}^\infty \frac{n x^n}{1 - \left( \frac{x}{1+x} \right)^n}$?
During a homework assignment, I ran into this series:
$$S(x) = \sum_{n=0}^\infty \frac{n x^n}{1 - \left( \frac{x}{1+x} \right)^n}$$
Graphing $S(x)$ shows that it converges for some values of $x$, ...
- 225
1
vote
0
answers
69
views
Indefinite integral of elliptic integrals
Derivative of complete elliptic integrals, $E(k)$, $K(k)$, etc., are known. But, I don't know about their integrals.
I tried to evaluate the indefinite integral
$$\int_0^k K(k)dk\tag1$$
and ended up ...
- 11.9k