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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".

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closed-form solution of a system of nonlinear trigonometric equations converting Euler angles to spherical

Following this and this questions I want to solve the system of nonlinear trigonometrical equations: $$ \displaystyle \left[\begin{matrix} \sin{\phi} \sin{\gamma} = \sin{\left(\alpha \right)} \cos{\...
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Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
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Closed form expression of a recursive relation

Consider the function $f \colon (0, +\infty) \rightarrow \mathbb{R} \colon x \mapsto 1/2(1 - e^{-1/x})$. Consider the recursive relation $$ \left \{ \begin{array}{ll} g_0(x) = f(x) & \\ g_k(x) = ...
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More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$

I. In this post, the OP asks about the particular log sine integral, $$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
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Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?

(This is related to this question.) Define the integral, $$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$ with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$, $$...
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1answer
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Finding closed form for a product

I have no idea how to solve the below product whose closed form I need to solve a problem, can anyone at the very least guide me to a solution or give me a source to check? $\prod_{k=1}^{n}\dfrac{2^k-...
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The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...
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Radical Equation with Symmetry

Is there an approach to solve the following radical equation in $x$? $(P - 1)x = QR\left( Q + \sigma \sqrt{Q^2 + x} \right)\left( R + sign(P-1) \sigma \sqrt{R^2 + x} \right)$ We know $x\in (0,1)...
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2answers
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Solving Recursion With Characteristic Polynomial

Say a linear recursion has the form: $a_n=a_{n-1}+a_{n-2}+2^{n-2}$, I know it would be possible to solve it with $a_n=a_{n-1}+a_{n-2}$ and $a_n=2^{n-2}$ and then combine the two using some coefficient ...
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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 ...
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1answer
90 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)\...
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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 ...
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1answer
51 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}\...
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1answer
105 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 $n>0$ ?
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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 + {\...
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2answers
56 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 ...
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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\...
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1answer
44 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. ...
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1answer
30 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 ...
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1answer
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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 ...
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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 ...
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4answers
116 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(...
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0answers
55 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{\...
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1answer
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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 ...
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2answers
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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 =\...
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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^...
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3answers
115 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: $$\...
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2answers
118 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'...
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3answers
246 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 ...
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1answer
92 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+...
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1answer
51 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, $A$, $B$, $C$ ...
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3answers
153 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\...
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1answer
25 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 ...
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0answers
93 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\...
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3answers
147 views

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

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!
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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 ...
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3answers
113 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 ...
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4answers
254 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 ...
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0answers
55 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=\...
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2answers
256 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 ...
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0answers
21 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,\...
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0answers
17 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
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4answers
219 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 ...
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2answers
50 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}? \...
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2answers
74 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 ...
1
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1answer
53 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
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1answer
81 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
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1answer
71 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 ...
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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\...
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1answer
95 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 ...