A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
30
votes
0answers
314 views
+500
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:
...
26
votes
3answers
337 views
+500
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 ...
25
votes
5answers
336 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?
$$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
22
votes
2answers
1k views
Closed form for improper definite integral involving trig functions and exponentials?
Is it possible to calculate the following definite integral in a closed form?
$$ \int_0^\infty \left| \sin x \cdot \sin (\pi x) \right| e^{-x} \, dx$$
21
votes
2answers
204 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:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
20
votes
3answers
259 views
+100
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
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
18
votes
1answer
158 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.
17
votes
3answers
967 views
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
16
votes
2answers
1k views
What does closed form solution usually mean?
This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form ...
16
votes
1answer
178 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 ...
16
votes
4answers
205 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:
$$\int\frac{\cos x}{\sin ...
15
votes
5answers
769 views
Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$
Today, we had a math class, where we had to show, that $a_{100} > 14$ for
$$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$
Apart from this task, I asked my self: Is there a closed form for this ...
15
votes
2answers
213 views
Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$
Here is another infinite sum I need you help with:
$$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$
I was told it could be represented in terms of elementary functions and integers.
15
votes
1answer
149 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
14
votes
2answers
273 views
Who has the upper hand in a generalized game of Risk?
So, I played a game of Risk the other day for the first time since I was very little. I was frustrated to discover that I couldn't compute (at least not in my head) whether the attacker or the ...
13
votes
3answers
158 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
13
votes
1answer
102 views
Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$?
Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$, where $\mu(n)$ is the Möbius $\mu$-function?
13
votes
1answer
95 views
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
I encountered this integral in my calculations:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
11
votes
3answers
498 views
closed form of $\sum \frac{1}{z^3 - n^3}$
I am currently trying to find a closed form expression for $\displaystyle f(z) = \sum_{n \in \mathbb{Z}} \frac{1}{z^3 - n^3}$, $z \in \mathbb{C}$. After a set of twists and turns, I have hit a wall.
...
11
votes
2answers
187 views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
11
votes
0answers
66 views
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
10
votes
4answers
120 views
Closed form for $n$th derivative of exponential of $f$
What is the closed form for:
$$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
10
votes
3answers
114 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 $\log x$?
10
votes
1answer
409 views
Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
Most of us are aware of the classic Gaussian Integral
$$\int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}$$
I would be interested in evaluating the similar sum
$$\sum_{x=0}^\infty e^{-x^2}$$
Now, ...
10
votes
1answer
116 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. ...
10
votes
1answer
102 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 ...
9
votes
3answers
109 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)\ $$e^{-x}\,\mathrm dx.$
9
votes
3answers
110 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
9
votes
1answer
81 views
Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right) e^{- x} d x$ have a closed-form expression?
Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right)
e^{- x} d x$ (with 6 variables) where W is the Lambert W function (also known as ProductLog in Mathematica) have a ...
9
votes
1answer
107 views
Closed form for $\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$
Please help me to find a closed form for this integral:
$$\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$$
9
votes
1answer
69 views
Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$
Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
9
votes
2answers
94 views
$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
8
votes
1answer
93 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
7
votes
1answer
236 views
How can I find $\sum\limits_{n=0}^{\infty}\left(\frac{(-1)^n}{2n+1}\sum\limits_{k=0}^{2n}\frac{1}{2n+4k+3}\right)$?
prove that$$\sum_{n=0}^{\infty}\left(\frac{(-1)^n}{2n+1}\sum_{k=0}^{2n}\frac{1}{2n+4k+3}\right)=\frac{3\pi}{8}\log(\frac{1+\sqrt5}{2})-\frac{\pi}{16}\log5 $$
This problem, I think use
...
7
votes
1answer
57 views
Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$
I need help with solving this integral:
$$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$
where $\text{Li}_{s}(z)$ is the polylogarithm.
7
votes
1answer
94 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
7
votes
1answer
74 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$
Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
This is a ...
7
votes
0answers
157 views
Addition formula for $f_n(x+y)$ in closed form.
$n$ is a positive integer.
$$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$
$f_n(0)=0$,
$f_n'(0)=1$ then
I am looking for the addition formula for $f_n(x+y)$ in closed form.
if $n=1$ then
...
6
votes
4answers
123 views
closed form for $d(4)=2$, $d(n+1)=d(n)+n-1$?
I am helping a friend in his last year of high school with his math class. They are studying recurrences and proof by inference. One of the exercises was simply "How many diagonals does a regular ...
6
votes
3answers
183 views
Unforeseen issue in my MastersThesis: Is there a “closed form” solution?
I'm an electrical engineer and I recently came across an unforeseen issue in my masters thesis because I lack a deeper mathematical education.
I want to know for which positive real $x$ the following ...
5
votes
3answers
157 views
Calculate $\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $
Is any formula which can help me to calculate directly the following sum :
$$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$
Or can you help me for an approximation?
Thanks :)
5
votes
3answers
148 views
Closed form for the sum of even fibonacci numbers?
I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for
$$
\sum_{i=1}^n F_{3i}
$$
...
5
votes
2answers
73 views
How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$
I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits:
...
5
votes
2answers
93 views
Is there a closed form expression for the first half of the Binomial series?
I'm looking for a closed form expression for the sum
$P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$,
where $n$ is a given positive integer and $k$ runs over nonnegative integers between $0$ and ...
5
votes
1answer
137 views
Tough Inverse Fourier Transform
In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
5
votes
0answers
53 views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\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}}.$$
It can be represented as
...
4
votes
5answers
184 views
What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$?
What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for ...
4
votes
2answers
74 views
Closed form for $\sum^{\infty}_{{i=n}}ix^{i-1}$
How can I find a closed form for:
$$\sum^{\infty}_{{i=n}}ix^{i-1}$$
It looks like that's something to do with the derivative
4
votes
3answers
325 views
How do I find if $\frac{e^x}{x^3} = 2x + 1$ has an algebraic solution?
Is there some way of solving $$\frac{e^x}{x^3} = 2x + 1 $$
non-numerically?
How would I go about proving if there exists a closed form solution? Similarly how would I go about proving if there exists ...
4
votes
4answers
479 views
Closed form solution of Fibonacci-like sequence
Could someone please tell me the closed form solution of the equation below.
$$F(n) = 2F(n-1) + 2F(n-2)$$
$$F(1) = 1$$
$$F(2) = 3$$
Is there any way it can be easily deduced if the closed form ...
