A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
3
votes
0answers
34 views
Closed form of specific series
I'm working on a problem that involves the integrals of various Bessel functions that Mathematica can't symbolically handle. I've managed to grind out the transformations and integrals by hand, and ...
0
votes
0answers
49 views
Finding closed form for the given recurrance
I am looking for a closed form for the the following recurrance:
$$
D(z,k,l) = \begin{cases}
(z-l) D(z-1,z,z-2-l) + (l+1) D(z-1,z,z-3-l), & k = z+1, \\
(z-l) D(z-1,k-2,l) + (l+1) D(z-1,k-2,l+1), ...
3
votes
3answers
70 views
The value of the $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}$
What is the value of $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}?$$
2
votes
0answers
28 views
Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,du$
Does the following integral have a closed form in terms of known functions?
$$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,du.$$
The parameters are possibly complex, and satisfy
$$\Re(c)>-1, ...
5
votes
1answer
57 views
Limit of an integral related to the beta function: $\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}$.
Consider the following limit:
$$ Z(\beta) = \lim_{z\to1-}\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}. $$
(This is related to this question.)
What is the closed form for this limit?
...
2
votes
2answers
68 views
Closed form of $\sum\limits_{i=1}^n\left\lfloor\frac{n}{i}\right\rfloor^2$?
Does $\displaystyle\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor^2$ admit a closed form expression?
11
votes
2answers
145 views
A Challenging Euler Sum $\sum\limits_{n=1}^\infty \frac{H_n}{\tbinom{2n}{n}}$
Recently, I encountered a strange series involving Harmonic Numbers and Binomial Coefficients both.
According to Mathematica:
$$\displaystyle \sum_{n=1}^\infty \frac{H_n}{\binom{2n}{n}} = ...
8
votes
2answers
111 views
How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?
I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$,
where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$.
Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
1
vote
1answer
38 views
Estimate the scale of the power series with Poisson pdf-like terms
Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue.
I would like to have an estimate for the series
$$P(t) = ...
12
votes
1answer
113 views
What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$
Let $\operatorname{li} x$ denote the logarithmic integral:
$$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$
Is it possible to find a closed form of the following integral?
$$\int_0^1\ln(-\ln x) ...
13
votes
1answer
90 views
Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$
Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits:
...
10
votes
1answer
129 views
How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?
I am interested in finding a general formula for the following integral:
$$\int_0^\infty J_\nu(x)^3dx,\tag1$$
where $J_\nu(x)$ is the Bessel function of the first kind:
$$J_\nu(x)=\sum ...
3
votes
1answer
41 views
Sylvester's sequence: is there an exact closed form?
I'm afraid this is one of those "amateur mathematician with no journal access" questions. Anyhow, Wikipedia (here) and OEIS give this closed form for the terms Sylvester's sequence:
$S_n = \lfloor ...
6
votes
2answers
71 views
Is there a simpler closed form for $\sum_{n=1}^\infty\frac{(2n-1)!!\ (2n+1)!!}{4^n\ (n+2)\ (n+2)!^2}$
I have the following infinite sum that can be expressed in terms of the generalized hypergeometric function:
$$\sum_{n=1}^\infty\frac{(2n-1)!!\ (2n+1)!!}{4^n\ (n+2)\ ...
3
votes
4answers
101 views
How to compute $\int^{\infty}_{0} t^{(\frac1n-1)}\cos t \,\mathrm{d}t$?
How to calculate the below integral?
$$
\int^{\infty}_{0} \frac{\cos t}{t^{1-\frac{1}{n}}} \textrm{d}t = \frac{\pi}{2\sin(\frac{\pi}{2n})\Gamma(1-\frac{1}{n})}
$$
where $n\in \mathbb{N}$.
14
votes
1answer
136 views
How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
How can I prove the following conjectured identity?
...
5
votes
2answers
157 views
How to prove a generalized integral identity
$$
\int_{0}^{\infty }\frac{t}{(e^{2\pi t}-1)(1+t^{2})}dt=-\frac{1}{4}+\frac{\gamma}{2}
$$
where $\gamma$ = Euler Gamma
$$
\int_{0}^{\infty }\frac{t}{( e^{2\pi t}-1)(1+t^{2}) ^{2}}dt=\frac{\pi^2}{24} ...
3
votes
1answer
70 views
Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$
For certain values of x and y, the sum $$\sum_{i=1}^{\infty}{\frac{1}{x^i-y^i}}$$ converges...is there a way to get the exact value, given x and y?
8
votes
2answers
129 views
How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$
I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits:
...
11
votes
1answer
73 views
Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$
I need help with solving this integral:
$$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$
where $\text{Li}_{s}(z)$ is the polylogarithm.
9
votes
1answer
128 views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}.$$
It can be represented as
...
1
vote
2answers
31 views
Closed form for Exponential Conditional Expected Value & Variance
I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution.
For example:
$$ E(X|x > a) $$ where X is exponential with mean ...
11
votes
1answer
154 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
4
votes
1answer
78 views
Closed form for the series $\sum\limits_{n=0}^\infty \frac{\exp(\cos(n))}{n!}$
Does anyone have any ideas on how to find a closed form for the following expression? It comes up when trying to bound a particular integral. The sum is:
$$\sum_{n=0}^{\infty} ...
11
votes
3answers
151 views
How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?
I need some help with solving this integral involving Bessel function:
$\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
10
votes
5answers
166 views
Closed form for $n$th derivative of exponential of $f$
What is the closed form for:
$$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
1
vote
2answers
77 views
Summation involving subfactorial function
Inspired by this post:
Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
16
votes
1answer
132 views
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
I encountered this integral in my calculations:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
10
votes
3answers
138 views
Closed form for n-th anti-derivative of $\log x$
Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
42
votes
1answer
695 views
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
...
4
votes
2answers
113 views
Closed form for $n$-th derivative of exponential
I need the closed-form for the $n$-th derivative ($n\geq0 $):
$$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$
Thanks!
By following the suggestion of Hermite polynomials:
...
4
votes
2answers
81 views
Closed form for $\sum^{\infty}_{{i=n}}ix^{i-1}$
How can I find a closed form for:
$$\sum^{\infty}_{{i=n}}ix^{i-1}$$
It looks like that's something to do with the derivative
15
votes
2answers
157 views
Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$
I need to find a closed form for these nested definite integrals:
$$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$
The inner integral can be ...
10
votes
2answers
132 views
$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
15
votes
3answers
185 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
23
votes
2answers
245 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
27
votes
3answers
405 views
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
2
votes
3answers
73 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$
$m$ is a nonzero positive integer, and $b$, ...
11
votes
1answer
111 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
19
votes
4answers
231 views
$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin ...
26
votes
3answers
427 views
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
I need to calculate the following integral:
$$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$
where
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
17
votes
2answers
239 views
Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$
Here is another infinite sum I need you help with:
$$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$
I was told it could be represented in terms of elementary functions and integers.
15
votes
1answer
194 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
2
votes
0answers
71 views
A photon in expanding Universe (a snail on a tree)
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
3
votes
3answers
50 views
Closed form of a recurrence relation using generating functions
It's been awhile since I have done this. The sequence is $\displaystyle a_n = a_{n-1} + 5~a_{n-2}$ with $a_{0}=0$ and $a_{1}=1$.
I found the generating function to be $\displaystyle G(x) = ...
22
votes
1answer
185 views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$
Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
8
votes
1answer
105 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
28
votes
5answers
449 views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$
Is there a closed form for the following infinite product?
$$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
18
votes
1answer
217 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
0
votes
1answer
52 views
Prove a non-empty subset is closed in an inner product space
I hope someone would be able to help me with the finer details of this proof.
Problem:
M is a non-empty set in an Inner Product Space (IPS) X.
I need to show that the annihilator of M which is ...
