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
votes
2answers
149 views

Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
2
votes
1answer
107 views

Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

I am trying to evaluate $$P=\frac\pi2\sum_{n\geq1}\frac{{2n\choose n}}{4^n n^2}$$ I used the beta function to show that $$P=\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$$ IBP: $$P=\sin^{-1}(x)\...
0
votes
2answers
35 views

Solving a recursive relation

Let $\{c_t\}_{t = 1}^k$ be a (non-monotone) sequence of real numbers such that $c_t \in (0, 1]$ for all $t = 1, \dots, k$. Consider the recursive sequence $$ \left \{ \begin{array}{ll} x_1 & = c_1 ...
1
vote
1answer
55 views

Closed form for $I(a)=\int_0^\infty \ln\left(\tanh(ax)\right)dx$?

I have been messing around with this integral that has some particular special values$$I(a)=\int_0^\infty \ln\left(\tanh(ax)\right)dx$$ I found that $$I(1)=-\frac{\pi^2}{8}$$ $$I\left(\frac{1}{2}\...
1
vote
1answer
123 views

Closed-form solution for $\int_{0}^{\pi/4} e^{-(n^2\sec^2x)/2}\,dx$

Is there a closed form solution for the following integral $$\int_{0}^{\pi/4} e^{-(n^2\sec^2x)/2}\,dx$$ for $n>0$ ?
14
votes
4answers
485 views

How to find $\int_0^{{\pi}/{2}} (\pi x-4x^2)\log(1+\tan x)\mathrm dx$

How do you find the integral of $$ \int_0^{\pi/2}\left(\pi x - 4x^{2}\right) \log\left(1 + \tan\left(x\right)\right)\,\mathrm{d}x $$ The integral can be simplified to $$ \int_0^{\pi/2}x\left(4x + {\...
2
votes
2answers
67 views

Generalized central binomial coefficients convolution

It is well-known that \begin{align*} \sum_{i=0}^n \binom{2i}{i}\binom{2n-2i}{n-i} = 4^n, \end{align*} where one might use combinatorial arguments or generating function technique to prove this. Now I ...
1
vote
0answers
67 views

Closed form for the sum?

Is there a closed form for this sum? It's a mixing summation of different terms in the zeta function with different values of $s.$ $$ S=\frac{1}{1^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{4^5}+ \cdot\...
1
vote
1answer
50 views

Evaluating a particular sum of products of exponential functions

I'm trying to get a closed-form expression for a particular type of sum, or at least a good way to approximate such sums numerically. I've tried using Mathematica, Maxima, etc, to no avail so far. ...
0
votes
1answer
32 views

Exact formula of the roots of a polynomial

I'm looking for a closed formula given one (or all) root of a polynomial $P=aX^4 +bX^3+cX^2+dX+e$. I'm not interested in the efficiency of such a formula. On the contrary, I would like to show my ...
1
vote
1answer
93 views

Is there a closed form, or cleaner way of writing a function satisfying $\frac{d^nf(x)}{dx^n}|_{x=0}=f(n)$ for all $n$?

Given the following, and assuming that $f(x)$ is infinitely differentiable: $$\frac{d^nf(x)}{dx^n}\Bigg|_{x=0}=f(n)$$ What functions $f$ could satisfy this equation? Do any functions of $f$ have a ...
7
votes
0answers
134 views

The sine cardinal function and $F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 0$

Define the function, $$F_n=\frac12-\int_0^\infty \frac{\sin^n x}{x^n}\,dx+\sum_{x=1}^\infty \frac{\sin^n x}{x^n}\tag1$$ where $\rm{sinc}^n(x)=\frac{\sin^n x}{x^n}$ is the sine cardinal function. We ...
3
votes
4answers
127 views

How can I compute this integral in closed form : $\int_0^{\frac{π}{4}}\ln^2(\tan x)dx$

How can I compute this integral in closed form : $\displaystyle\int_0^{\displaystyle \tfrac{π}{4}}\ln^2(\tan x)dx$ How can use Fourier series here ? $-2\displaystyle \sum_{n=0}^{\infty}\frac{\cos(...
0
votes
0answers
57 views

Evaluating the integral $\int_0^{\pi/2}\frac{\cos^{3/2}x}{\sin x\sqrt{\lambda-2\cos x-2\ln(1-\cos x)+2\ln(\sin x)}}\,\mathrm dx$

I'm trying to find this indefinite integral $$\int\frac{\cos^{3/2} x}{\sin x\sqrt{(\lambda-2\cos x-2\ln(1-\cos x)+2\ln(\sin x))}} \,\mathrm dx$$ Or $$\int_0^{\pi/2}\frac{\cos^{3/2}x}{\sin x\sqrt{\...
9
votes
1answer
62 views

Methods to derive a closed form for $I_n=\int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}dx$

I've stumbled onto this general integral that has closed form values for the $n\in \Bbb{Z^+}$ $$I_n=\int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}dx$$ Obviously $I_0=0$ but higher values of $n$ yield ...
0
votes
2answers
98 views

Integral $\int_0^{\pi / 4}\frac{x\tan^2 x\ln\left(\tan x\right)}{\cos^2 x}dx$

Compute in closed form without using series: $$I =\int_0^{\pi / 4}\frac{x\tan^2 x\ln\left(\tan x\right)}{\cos^2 x}dx$$ I thought of using: $y=\tan x$ then $dy=\frac{1}{\cos^2 x}$, so : $$I =\...
-1
votes
2answers
82 views

Compute in closed form : $\int_0^{\frac{π}{4}} x\ln(\tan x)\left(1-\frac{1}{\cos^2 x}\right)dx$

Question : Compute in closed form without use series $I =\displaystyle\int_0^{\pi / 4} x\ln\left(\tan x\right)\left(1-\frac{1}{\cos^2 x}\right)\,dx$ I think use : $y=\tan x$ then $dy=\frac{1}{\cos^...
2
votes
3answers
125 views

Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$

I am trying to find closed form for this integral: $$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$ Where $a>0$. My try: Let: $$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$ Then: $$\...
5
votes
2answers
127 views

Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Compute the following integration without harmonic series or Fourier series : $I=\displaystyle\int_0^{\frac{π}{4}}x\ln(\tan x)dx$ Wolfram alpha give $I=\frac{7\zeta(3)-4πC}{16}$ Where $C$ : Catalan'...
12
votes
4answers
319 views

Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

Prove that$$I=\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx=\frac{\pi}{3}\ln(2+\sqrt 3)+\frac43G$$ I've found this integral in my notebook and perhaps I encountered it before since it looks quite ...
0
votes
1answer
94 views

Compute in closed form that $S=\sum_{n=1}^{\infty}\frac{1}{6n^5+15n^4+10n^3-n}$

Compute the following sum : S=$\sum_{n=1}^{\infty}\frac{1}{6n^5+15n^4+10n^3-n}$ My attempt : Use partial fraction : $6n^5+15n^4+10n^3-n=n(n+1)(2n+1)(3n^2+3n-1)$ $S=\sum_{n=1}^{\infty}(\frac{9(2n+...
0
votes
1answer
52 views

How to solve this integration or does this have a closed form? [closed]

The integral I am dealing with is below. I need to find the closed-form expression of this integral. $$\int_0^\infty \ln\left(1+\frac{A}{1+B+Cx}\right)\frac{e^{-x/M}}{M}\,dx.$$ Here, $A$, $B$, $C$ ...
8
votes
3answers
160 views

What is $\sum_{k=1}^\infty \rm{sinc}^8(k)$ using the sine cardinal function?

Given the sine cardinal function, $$\rm{sinc}(x) = \frac{\sin x}x$$ for $x\neq0$. We have the nice evaluations, $$\sum_{k=1}^\infty \rm{sinc}(k) = \sum_{k=1}^\infty \rm{sinc}^2(k)=-\tfrac12+\tfrac12\...
0
votes
1answer
29 views

How do I find a closed form expression for a sum

If $k$ and $n$ are positive integers, how do I give a simple closed form expression for the sum $\sum_{a_1+···+a_k=n} {n \choose a_1,...,a_k}$ I'm not sure the process of finding a closed form ...
4
votes
2answers
184 views

How to find $\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}$ using real analysis and in an elegant way?

I have already evaluated this sum: \begin{equation*} \sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}=4\operatorname{Li_4}\left( \frac12\right)+\frac{13}{8}\zeta(4)+\frac72\ln2\zeta(3)-\ln^22\zeta(2)+\frac16\...
4
votes
3answers
153 views

Find : $\int_0^{\infty}\frac{\cos (2ax)}{x}\tanh (2πx)dx$ [closed]

I'm try to Find : $$\int_0^{\infty}\frac{\cos (2ax)}{x}\tanh (2πx)dx$$ I don't have any idea to compute this type of integration Thanks!
0
votes
1answer
29 views

Binomial distribution with nonlinear function of successes

Is there a closed form expression for the following expression: $$\sum_{j=1}^{N-1} {N-1\choose j} q^j (1-q)^{N-1-j} \frac{c-jd}{e+jd}$$ where $c$, $e$, and $d$ are some real numbers? I wonder if the ...
1
vote
3answers
120 views

Finding $ \int_0^1\frac{\ln(1+x)\ln(1-x)}{1+x}dx$ [duplicate]

Calculate $$\int_0^1\frac{\ln(1+x)\ln(1-x)}{1+x}\,dx$$ My try : Let : $$I(a,b)=\int_0^1\frac{\ln(1-ax)\ln(1+bx)}{1+x}\,dx$$ Then compute $\frac{d^2 I(a,b)}{dadb}$. I'm happy to see ideas in ...
1
vote
4answers
257 views

Which of these claims are true?

Consider the sequences of numbers $\left\{0, 1, 2\right\}$ with length $n$. There are $3^n$ such sequences. I define each sequence like a function. If a function consists of {0,1,2} elements of the ...
1
vote
0answers
59 views

Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
11
votes
2answers
279 views

Integral $\int_0^1 \frac{\ln(1+x)}{1+x^3}dx$

Earlier today I saw this integral around here and gave it a try without success, unfortunately it got taken down so it didn't receive to much attention, but I think it's a nice integral (although it ...
0
votes
0answers
22 views

How to find: $f^{\alpha}_n(x)=1+\sum_{k=1}^{\infty} \frac{\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$

I am looking for a closed-form solution to $$f^{\alpha}_n(x)=a_0+\sum_{k=1}^{\infty} \frac{a_k\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$$ where we can take $a_k = 1, \forall k\in [0,\...
2
votes
0answers
18 views

Closed form solution for the difference of two poisson processes

I'm interested in whether there is a closed-form distribution of the time it takes two Poisson processes to output counts to have a fixed difference. For example, let $k_1$ ~ Poisson($\lambda_1t$) $...
4
votes
4answers
223 views

Closed form of recurrent arithmetic series summation

Knowing that $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ how can I get closed form formula for $$\sum_{i=1}^n \sum_{j=1}^i j$$ or $$\sum_{i=1}^n \sum_{j=1}^i \sum_{k=1}^j k$$ or any x times neasted ...
1
vote
2answers
51 views

Sum of alternating binomial-coefficient-type series

Let $D,n\in \mathbb N$ with $0<D<n$, and $y>0$ is a real number. Question: Is there a closed-form for the following alternating sequence \begin{equation} \sum_{k=0}^D (-y)^k {n\choose k}? \...
1
vote
2answers
79 views

Closed form for $\sum_{j=0}^{n}a^{-j}B_{j}B_{n-j}{n \choose n-j}$ [closed]

$n=2k+1$, $k\ge1$ Where $B_n$ ; Bernoulli number $$\sum_{j=0}^{n}2^{-j}B_{j}B_{n-j}{n \choose n-j}=-\frac{2^{n-2}+1}{2^n}\cdot nB_{n-1}\tag1$$ We manage to figure the closed form for $(1)$ We are ...
2
votes
1answer
67 views

Closed form solution of an SDP [closed]

Given symmetric positive definite matrices $A$, $M_1$ and $M_2$, is there any closed form solution for the following convex problem in $X$? $$\begin{array}{ll} \text{maximize} & \mbox{tr}(AX)\\ \...
2
votes
1answer
82 views

Take two numbers x and y between 1 and 100. What’s the probability that x/y is an integer?

It was stated, as an inconsequential remark, in some lecture notes I was reading that if we are to choose two natural numbers in a certain interval and divide one by the other, that it is quite likely ...
2
votes
1answer
75 views

Closed form for this integral $\int_0^\infty \frac{e^{-x} dx}{\sqrt{(x+a)^2+b^2}}$

This is a Laplace transform, however I couldn't find it in the tables and Wolfram doesn't know the answer either: $$I(a,b)=\int_0^\infty \frac{e^{-x} dx}{\sqrt{(x+a)^2+b^2}}$$ Some kind of closed ...
2
votes
3answers
96 views

Does this double product equal an exponential function?

I was looking at the graph of $$\prod_{n=1}^\infty\frac{\left(\Gamma(n+1)\right)^2}{\Gamma\left(n+x+1\right)\Gamma\left(n-x+1\right)}=\prod_{n=1}^\infty\prod_{k=1}^\infty\left(1-\frac{x^2}{\left(n+k\...
2
votes
1answer
96 views

Are these statements always true?

I haven't found an answer in my books. Although the question seems very simple, I want to ask. Are these statements always true? a) For any infinity non-negative integer sequence, if there is an ...
0
votes
3answers
81 views

Solving a modified norm minimization problem

I have the following minimization problem in $x \in \mathbb{R}^n$ $$\begin{array}{ll} \text{minimize} & \|x\|_2 - c^T x\\ \text{subject to} & Ax = b\end{array}$$ where $A \in \mathbb{R}^{m \...
4
votes
1answer
62 views

Finding a closed form for coefficients in $x^{3n}=x_0\left(a_nx+b_n+\frac {c_n}{x}\right)$

Consider, $$ x^3=x+1 $$ Let $x_0$ be a solution to the above equation. Now consider $x^{3n}$. For $n=2$ we have: $$ x^6=(x+1)^2 $$ $$ =x^2+2x+1 $$ $$ =x\left(x+2+\frac {1}{x}\right) $$ $$ =x_0\left(x+...
-2
votes
1answer
39 views

How to find the closed form of the summation below without changing the lower and upper bound of summation? [closed]

The lower bound of summation is i=0 and the upper bound of summation is log(n) - 1 (log is base 2).
1
vote
1answer
64 views

Arc length of $x^3 \sqrt{9-x}$ on $[0,9]$

This is supposed to be part of a student's Calc 2 homework; however, this seems to be an extremely difficult integration, and I couldn't figure it out. Find the arc length of $x^3 \sqrt{9-x}$ on the ...
0
votes
0answers
15 views

Tricky system of differences

We have a sequence $q(1,k)=q(1,k+44)$ for $k\geqslant0$ with special conditions, which gives us $q(1,k)=q(1,k±22)$ $q(1,11)=q(1,11(2m+1))=12$ and the first terms are $$0,2,4,6,8,6,8,8,10,10,12,12$$ ...
0
votes
0answers
49 views

Closed form solution for differential equation

I would like to solve the following equation:  $$\bar{t} = \int_0^{\infty} t F(t) \, dt = \int_0^{\infty} e^{ -4 ms \int_0^t p(t') \, dt' } \, dt$$ Just focusing on the exponential section: $ ...
0
votes
0answers
37 views

Analytic Integration, or, closed-form expression for an integral

I have been trying to integrate this function analytically for quite some time. I have already used Taylor series expansion, and it is not helpful. So, I want to integrate f = $\cos{(K_4 + K_5 x + \pi ...
0
votes
0answers
57 views

Is there a general formula for $\int{\big(\frac {\arctan x}{x^2+1} \big )}^{\frac1k}dx$ , with $k$ is positive integer?

I'm interested to know if there is a general formual for $$ \int\left[\arctan\left(x\right) \over x^{2} + 1\right]^{1/k}\mathrm{d}x $$ with $k$ is positive integer may present integral of fraction ...
2
votes
0answers
139 views

Closed form of :$ \int_{-\infty}^{\infty}\arctan\left(e^{-x^2 \text{erf}(x)}\right)\,\arctan\left(e^{x^2\text{erf(x)}}\right)\,dx $

That is one of interesting integral that i have accrossed in my text book when i have tried to understand some thing related to distribution theory in the statistics context , The following integral ...