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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69 views

Help in simplifying this double summation

Can I express the following double summation $$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$ where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,i \in [1:n], j \in [1:m]\}$ while $G(.)$ is any function ...
3
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0answers
33 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
4
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0answers
63 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
6
votes
3answers
98 views

Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

While I've studied integrals involving polylogarithm functions I've observed that $$\int_0^1 \operatorname{Li}_p(x) \, dx \stackrel{?}{=} \sum_{k=2}^p(-1)^{p+k}\zeta(k)+(-1)^{p+1},\tag{1}$$ for any ...
0
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0answers
66 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ...
5
votes
2answers
111 views

Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
2
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0answers
46 views

How to calculate alternating Euler sum [closed]

In How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? get $$\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2=\frac{1}{12}(\pi^2\log2-4(\log 2)^3-9\zeta(3)),$$ Similar, how to evaluate the series, ...
3
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1answer
63 views

Closed-form of a special value dilogarithm identity

Let $c$ be the following. $$c = \frac{1+i\sqrt 3}{3}\operatorname{Li}_2\left(1-\frac{i\sqrt 3}{3}\right)+\operatorname{Li}_2\left(\frac 34 + \frac{i\sqrt 3}{4}\right) + ...
9
votes
1answer
163 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
6
votes
2answers
172 views

Closed-form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it. Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then ...
3
votes
2answers
95 views

Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$

Is there a closed-form of the following sequence? $$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$ where $_2F_1$ is the hypergeometric function and $n ...
1
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0answers
39 views

Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function

The reciprocal gamma function has the following Taylor series. $$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$ where the $a_k$ coefficient are given by the followint recursion. $a_1=1$, ...
10
votes
2answers
195 views

Closed-form of $\int_0^1 B_n(x)\psi(x+1)\,dx$

Is there a closed-form of the following integral? $$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$ where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function. The motivation of the ...
3
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1answer
136 views

Closed-form of $\int_0^{\pi/4} \sin(\sin(x)) \, dx$

Let $I(b)$ is the following integral $$I(b)=\int_0^b \sin(\sin(x)) \, dx.$$ There are some $b$ value for that we know a closed-form of $I(b)$ in term of Struve function $\mathbf{H}_n(x)$. For ...
36
votes
1answer
2k views

The Wicked Integral

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] ...
13
votes
3answers
322 views

Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left[\ln\frac{(1+\alpha)^{1+\alpha}}{\alpha^\alpha}\right]$

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by ...
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0answers
50 views

Sum of Bessel functions and integral of exponential of sine or cosine

I'm looking to simplify the following sum of Bessel functions of the first kind: $$\sum_{q=-\infty}^{\infty}\frac{(-)^{q}}{2q+1}e^{iq\theta}I_{q}(\alpha^{2})$$ Motivated by a related question and by ...
5
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0answers
65 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
7
votes
4answers
231 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
3
votes
2answers
98 views

Log integrals II

By considering the integral \begin{align} I_{\mu} = \int_{0}^{\pi/4} \sin(2\theta) \, \left( \cos(\theta) - \sin(\theta) \right)^{\mu} \, d\theta \end{align} derivatives can be taken with respect to ...
10
votes
1answer
182 views

What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?

Some time ago I asked How to find $\displaystyle{\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$. Thanks to great effort of several MSE users, we now know that \begin{align} \int_0^1\frac{\ln^3(1+x)\,\ln ...
0
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0answers
14 views

Developing closed-form from recursive definition

Here is a recursively-defined function where c, d ∈ N. T(n) =    c, if n = 0 d, if n = 1 2T(n − 1) − T(n − 2) + 1, if n > 1 Carry out the five steps for repeated substitution to prove a ...
8
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6answers
198 views

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

How do we find $$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$ In the shortest and easiest possible manner? I cannot think of anything good.
1
vote
2answers
46 views

Proving that a closed form is true for $n=k+1$ with induction

I need to prove that a closed form formula is true for n=k+1. I need to use mathematical induction and explain every step, but I'm getting lost on this one. I already found the closed form, and I made ...
4
votes
1answer
137 views

Closed form of $\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$

Is there a closed form for: $$\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$$ where: $$h(x)=(1-x)^{\alpha}(A-Bx)^{\frac{1}{\gamma}-\alpha}$$ and ...
4
votes
1answer
44 views

How can I find a closed form for the summation (i^2)(-1^i+1) systematically?

In one of my homeworks I was given the following sequence $1^2-2^2+3^2-4^2+\dots (-1)^{n+1}n^2$, and I'm supposed to find a closed form formula and prove that it works. Rewriting this as a sum gives ...
11
votes
1answer
290 views

Two integral involving logarithm and polylogarithm function

Evaluate the following integrals $$\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1 + x}{2} \right)\,dx\\ .\\ \int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)\,dx$$
7
votes
1answer
120 views

Evaluate $\int_{0}^{1} \frac{\left[\rm{Li}_2\left(\frac{1}{2} \right)-\rm{Li}_2\left(\frac{1 + x}{2}\right)\right]\ln( 1 - x)}{1 + x}\,dx$

$\def\Li{{\rm{Li}}}$How to evaluate the following integral$${\large\int_0^1} {\frac{{\left[ {\Li_2\left( {\frac{1}{2}} \right) - \Li_2\left( \frac{1 + x}{2} \right)} \right]\ln \left( {1 - x} ...
4
votes
3answers
138 views

Evaluate $\int_{0}^1 x^{p}(\log x)^q dx$

Evaluate $$\int_{0}^1 x^{p}(\log x)^q dx$$ for $p \in \mathbb{N}$ and $q \in \mathbb{N}$.
0
votes
1answer
69 views

Writing in closed form this nasty expansion

hi I have to write the following in closed form, $$a_1 + a_2 + a_3 + a_1 v_{2}v_{1}+ a_1v_{3}v_{1}+ a_2v_{3}v_{2}+a_2v_{1}v_{2}+ a_3v_{1}v_{3}+a_3v_{2}v_{3}$$ $$ \sum_{(i,j)\in \mathcal{S}} a_i(1+ ...
7
votes
1answer
130 views

Closed-form of $\sum_{k=1}^{\infty}\arctan(1/k^3)$

Wolfram said that $$\sum_{k=1}^{\infty}\arctan\left(\frac{1}{k^2}\right)=\arctan\left(\frac{1-\cot\left(\frac{\pi}{\sqrt 2}\right)\tanh\left(\frac{\pi}{\sqrt 2}\right)}{1+\cot\left(\frac{\pi}{\sqrt ...
10
votes
2answers
303 views

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
4
votes
0answers
73 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
2
votes
1answer
142 views

Closed-form of $\int_{a}^{b}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx$ for some $a<b$

In this question I asked to prove that $$\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}.$$ If we take a look at the plot of the integrand, then we could see some symmetry-property. ...
3
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0answers
30 views

How to work with a recursive function with 2 recursive instances?

In class, we figured out how to find the closed form of a recursive definition through the "basic 5 steps method". Example function T(n): If n = 1, T(1) = 1 If n > 1, T(n) = T(n-1)+1 Step 1: ...
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0answers
16 views

Integration : Green's Function in estimating displacement of non-prismatic beams

I'm working on a non-prismatic structure similar to that in Figure 3 of Page 10 (345) from an article entitled: "Green’s function for the deflection of non-prismatic simply supported beams by an ...
0
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0answers
22 views

Prove that the composition of two “closed form functions” is itself a “closed form function”?

So I have been given the definition of a "closed form function" that is a set of functions built inductively (mapping from and to the complex) starting with the fact that the constant functions ...
0
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1answer
55 views

Any way to get a Recursive function from its closed form?

For an exercise, the goal is to find a recursive definition for a certain function. the function itself is as follows: f(a,b) is the # of binary strings of length a and with b more 1s than 0s. eg: ...
-1
votes
1answer
51 views

Integral involving the error function

Is there a closed form solution to the integrals \begin{align} I_{c} &= \int_{0}^{\infty} \cos(a x) \, \operatorname{erf}(b x) \, dx \\ I_{s} &= \int_{0}^{\infty} \sin(a x) \, ...
12
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3answers
350 views

Integral $\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}}\operatorname d \!x$

Could you please help me with this integral? $$\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}} \operatorname d \!x$$ Update: user153012 posted a result given by a computer that contains scary Appel ...
0
votes
1answer
45 views

How to write in closed form this nasty expression?

I have something like $$ v_1 l_1+v_1 l_2+ v_2l_1+v_2l_2$$ and I am trying to write it in closed form as such, $$\sum_{j=1}\sum_{i=1}v_il_j$$ I know this is not right but I want something like that. ...
6
votes
1answer
213 views

Prove ${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits. ...
1
vote
0answers
39 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} ...
8
votes
1answer
599 views

A long nasty limit problem

Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$$ My professor gives this ...
0
votes
1answer
48 views

How to find this integral $I=\int_{0}^{1} t^{-1/2}\left(1-\frac{t}{1+4y}\right)^{-1/2-ix}dt$

Question: Find the closed form of $$I(x,y)=\int_{0}^{1}t^{-1/2}\left(1-\dfrac{t}{1+4y}\right)^{-1/2-ix}dt$$ where $i^2=-1$ I have used Wolfram Alpha but it can't help me out. How to ...
2
votes
3answers
53 views

How to derive the closed form of this recurrence?

For the recurrence, $T(n) = 3T(n-1)-2$, where $T(0)= 5$, I found the closed form to be $4\cdot 3^n +1$(with help of Wolfram Alpha). Now I am trying to figure it out for myself. So far, I have worked ...
20
votes
1answer
503 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
9
votes
1answer
202 views

Generalized FoxTrot Series $F(a,b,q,x) = \sum_{k=q}^{\infty} \dfrac {(-1)^{k+1} k^a}{k^b+x}$

The FoxTrot Series is defined as: $$F = \sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1}.$$ Using partial fraction decomposition we can show that $$F = \frac 13 \left[ 1 - \ln2 + ...
2
votes
2answers
47 views

Sums with squares of binomial coefficients multiplied by a polynomial

It has long been known that \begin{align} \sum_{n=0}^{m} \binom{m}{n}^{2} = \binom{2m}{m}. \end{align} What is being asked here are the closed forms for the binomial series \begin{align} S_{1} &= ...
2
votes
1answer
97 views

A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $

This is another 'arctanlog' series: $$ S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $$ Maybe differentiating with respect to some parameter could be of interest. What ...