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

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3
votes
3answers
147 views

Prove that $\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right)$

While I was working on this question, I've found that $$ I=\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right), $$ where ...
3
votes
1answer
48 views

What is the analytic representation of $\sum_{n=0}^\infty \frac{ a^nx^n}{n!}$?

I only know that a geometric series is $$\frac1{1-x}\qquad (|x|<1)$$ and this looks similar.
5
votes
0answers
110 views

Simplifying a certain polylogarithmic sum in two variables

This question is related to my previous question here. While tinkering around for a solution I found that the integral there can be reduced to the problem of solving the following basic logarithmic ...
7
votes
1answer
306 views

Closed-form of $\int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx$

I'm interesed in the following integral, for $a,b>0$: $$ \mathcal{I}(a,b) := \int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx $$ Mathematica could evaluate it in term of hypergeometric functions, but ...
12
votes
0answers
169 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
1
vote
1answer
176 views

How do I write a closed form expression for $\sum _{i=0}^{n-1}$ in terms of n?

I am given this:$$\sum _{i=1}^n a_i = n^2-n,a_0=4$$ How do I write a closed form expression for $$\sum _{i=0}^{n-1}$$in terms of n? I know that for $$\sum _{i=1}^{n-1}$$ the expression would be ...
0
votes
0answers
30 views

Closed form for a recursive equation that include the ceiling function

Can someone help me with finding the closed form of g(n) in terms of n, A, and B? g(n<0)=0 ; g(0)=0 ; g(1)=0 ; g(n) = A + g(n-1) - ceiling[g(n-1)/B] , n>=2 , A and B are Natural numbers, ...
3
votes
2answers
63 views

Sum of the series $\sum u_n$ where $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$

While I'm able to prove that the series $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$ converges, I don't see the trick to compute the value of its sum starting at $n=2$. Any clue on the ...
4
votes
1answer
112 views

What's the $n$-th derivative of $\ln(\sin(x))$?

I want to find the $n$-th derivative of $\ln(\sin x)$, i.e. $$ \frac{d^n\ln(\sin x)}{dx^n} $$ where $x\in (0,\pi/2)$ such that $\sin x>0$. To make the problem definitely, $x=\pi/4$ is assumed. In ...
0
votes
0answers
96 views

Integration with Log of error function (erf)

Can anybody help me evaluating the closed-form or an approximate form of $H(x) = \int P(x) \ln(P(x)) \Bbb dx$ where $P(x) = \frac{C(x)}{v\int C(x) \Bbb dx}$ and $C(x) = {\frac ...
11
votes
2answers
238 views

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different ...
1
vote
2answers
48 views

Closed form of a power series

Find the function that represents the following sum: $\sum\limits_{k=0} ^\infty \frac{(n^2)}{n!} x^n$. Can't find the function that represents this.
19
votes
2answers
266 views

Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

I'm interested in this integral: $$I=\int_0^1\ln^3\!\left(1+x+x^2\right)dx.\tag1$$ Can we prove that ...
6
votes
1answer
169 views

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after ...
1
vote
0answers
43 views

What is the solution of this recursion, that's defined in terms of a sum, but with this $1$ odd twist?

$$ F(n) = \sum_{i=0}^{n} F\left(\left\lfloor\frac{i}{5}\right\rfloor\right) $$ I encountered this odd looking functional equation, while perusing the site yesterday. I'd be interested in seeing a ...
4
votes
0answers
52 views

Can anyone identify the function that represents this infinite product?

$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful ...
1
vote
0answers
31 views

Closed form expressions for series solutions of D.E's

So, I was reading about power series solutions to differential equations and the author stressed how one should not bother trying to identify the final series solution with any closed form expression. ...
12
votes
2answers
222 views
4
votes
1answer
80 views

Summation of series containing logarithm: $\sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)}$

How do I find the sum of the series: $$\ln \frac{1}{4} + \sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)} $$ I tried expanding the terms on numerator and denominator and got $$\ln \frac{1}{4} + ...
7
votes
3answers
164 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
1
vote
1answer
60 views

Closed form of integral involving Bessel $J_0$

I am trying to find a closed form (in terms of known functions) for this integral: $$ \int_{0}^\Lambda\!\!\! \text{d} k\, J_0( k x)\sin (k y)$$ where $x>0$, $\Lambda>0$, $y\in \mathbb{R}$ I ...
8
votes
2answers
172 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
12
votes
1answer
361 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
0
votes
1answer
34 views

Minimum of sum of squares over sums

I am trying to minimize $\phi(\alpha)$, where $\alpha \in \mathbb{R}^K$. $\phi(\alpha) = \frac{R^2 + G^2 \gamma \sum_{i=0}^{K} A_i \alpha_i^2}{\sum_{i=0}^{K} A_i \alpha_i} $ Where, $A_i = \gamma ...
5
votes
2answers
695 views

Only once differentiable

Is there any example of a real function that is one-time-only differentiable, meaning there is $f'(x)$, but no $f''(x)$? I haven't been able to find any example... Of course it would be preferred if f ...
7
votes
3answers
172 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
13
votes
4answers
291 views

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

Let $H_q$ denote harmonic numbers (generalized to a non-integer index $q$): $$H_q=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+q}\right)=\int_0^1\frac{1-x^q}{1-x}dx=\gamma+\psi(q+1),\tag1$$ where ...
4
votes
2answers
85 views

Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form?

Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$, can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be ...
4
votes
1answer
78 views

closed-form of an integral similar to Bessel function

The integral form of the $n$-th modified Bessel function of the first kind is $$ I_n(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos\theta}\cos(n\theta)\;d\theta. $$ However, I found an integral $$ ...
27
votes
2answers
552 views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
7
votes
2answers
317 views
8
votes
3answers
234 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
2
votes
1answer
58 views

Computing the integral of $-1/f''$

I think this is a very silly question but I have some problems nonetheless. If I know that $g'=-\frac{1}{f''}$, is then $$ g=(f')^{-1}? $$
3
votes
1answer
114 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
1
vote
0answers
65 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
8
votes
4answers
197 views

Is there a formulaic way to go from $\sum_{k=1}^{n} \frac{1}{k}$ back to $n$?

Say you want to sum $g(n) = \sum_{k=1}^{n} \frac{1}{k} = L$. Is there a simple formula to go from $L$ and deduce $n$? My attempt: For $n = 1$, the formula is $L$. Assume there is a formula for all ...
8
votes
1answer
177 views

Closed-form of the hypergeometric function ${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{array}\middle|\,-t\right)$

Inspired by this question and by using Mathematica the following conjecture seems to be true for all nonzero complex $t$ number: ...
8
votes
1answer
95 views

$\sum_{n=1}^\infty \frac{1}{(n^2-1)!} - \sum_{n=1}^\infty \frac{1}{(7n+1)!}$ is almost $1+1/6$

I've recognized, that $$\mathcal{S} = \sum_{n=1}^\infty \frac{1}{(n^2-1)!} - \sum_{n=1}^\infty \frac{1}{(7n+1)!} \approx 1.1666666666666666666657785992648796$$ which is almost $1+1/6$. I think it is ...
6
votes
1answer
223 views

Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$

I've conjectured the following identity for $n\geq0$ integers: $$ \int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx = (-1)^{n+1}n! \cdot \left(-\operatorname{Ei}(1)+\sum_{k=1}^{n+1} ...
9
votes
1answer
244 views

Does this functional equation have a non-trivial closed form solution?

$$P(c \cdot x) = \cos(x) P(x)$$ For $c=2$, $P(x) = \sin(x)/x$ is a solution to this. I don't know if there's a closed-form solution for $c \ne 2$. Rather than add my own attempt at solution, which ...
-1
votes
1answer
76 views

Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it. The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) ...
1
vote
1answer
54 views

How to prove that a sequence with recurrence relation has no closed form expression?

It is always easy to forge recurrence relations. E.g. $$a_{n+1}=2a_n+\dfrac{1}{a_n}, a_0=1$$ But it is always hard to find the general closed form expression. And it is even harder to prove that ...
8
votes
1answer
194 views

Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$

Is there a direct way to evaluate the following series? $$ \sum_{n=1}^{\infty}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)=\frac12\ln^2 2. \tag1 $$ I've tried telescoping ...
3
votes
0answers
79 views

Infinite Series $\sum_{m=1}^{\infty} \frac{(-1)^{m-1}}{m^m}$

Any ideas to calculate this infinite sum? The ratio test guarantees the convergence; $$\lim_{m\to \infty} \frac{m^m}{(m+1)^{(m+1)}}=0<1$$
5
votes
2answers
161 views

Closed form for $(2^1-1)(2^2-1)…(2^k-1)$?

Is there closed form for $\prod_1^{i=k}(2^i-1)$ ? I found that it is the product of the terms of the following arithmetico-geometric sequence : $$\{u_1=1,u_{n+1}=2u_n+1\}$$ I found nothing with ...
1
vote
3answers
102 views

A nifty series involving $\cosh(x)$

There are many series that can be found in the literature that are entertaining. Here is yet another. What is the resulting value of the series $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \, ...
3
votes
0answers
116 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
8
votes
2answers
157 views

Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$

How could we prove that $${\LARGE\prod_{\Large n\ge0}}~\frac{\cosh\left(n^2+n+\dfrac12\right)+i\sinh\left(n+\dfrac12\right)}{\cosh\left(n^2+n+\dfrac12\right)-i\sinh\left(n+\dfrac12\right)}~=~i$$ ...
5
votes
0answers
314 views

Challenging integral: $\int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx$

I'd like to find a symbolic form for the following integral: $$ f(\alpha, Z) = \int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx $$ It is given that $0 \le \alpha \le 1$ and $0 \le Z < 1$. The following ...
2
votes
3answers
403 views

Generating Functions and closed form [closed]

I read somewhere that we can use generating functions to find closed form of a sequence. So what is the difference between a generating function and closed form of a sqeunce?