Linked Questions

131 votes
10 answers
29k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: $$\begin{...
79 votes
8 answers
9k views

Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the ...
32 votes
5 answers
2k views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
user avatar
36 votes
6 answers
2k views

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

One of the possible ways of computing the series is to obtain the generating function, but this might be a tedious, hard work, pretty hard to obtain. What would you propose then? $$\sum_{n=1}^{\...
  • 43.4k
36 votes
2 answers
3k views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
8 votes
3 answers
2k views

Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$

Evaluate $\displaystyle \int\limits_0^1 \dfrac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$ I was wondering if the above had some kind of a closed form, here some of the special cases have ...
12 votes
2 answers
345 views

Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?

(This summarizes my posts on Nielsen polylogs.) I. Question 1: How to complete the table below? Consider the special cases $z=-1$ and $z=\frac12$. Given the Nielsen generalized polylogarithm, $$S_{n,...
10 votes
2 answers
327 views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^5 2^n}$

Given the nth harmonic number $ H_n = \sum_{j=1}^{n} \frac{1}{j}$, we get from this post that apparently, $$\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n= S_{k-1,2}(z) + \rm{Li}_{\,k+1}(z)$$ for $-1\leq z\...
4 votes
2 answers
173 views

How to evaluate $\int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{dx}{x}$

$$I=\int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{\mathrm dx}{x}$$ $\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$ $$\int_{0}^{\infty}\left(1-\frac{x}{2}+\frac{x^2}{3}+\cdots\...
user avatar
4 votes
1 answer
228 views

Interesting Logarithmic Integral: $\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx $ [closed]

Other than numerical approximation, how can we calculate the closed form of this integral? $$\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx $$
6 votes
2 answers
136 views

On a certain integral that involves a product of powers of logarithms.

This is a follow-up question to the following questions: Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$ Closed form for ${\large\int}_0^1\frac{\ln^4(1+x)\ln x}x \, dx$...
  • 9,303
4 votes
2 answers
122 views

The closed form representations of Integrals of logarithm functions

I wish to find a closed form representations of the following integral $$\int\limits_{0}^1\frac{\log^p(x)\log^r\left(\frac{1-x}{1+x}\right)}{x}dx=?$$ Here $p\ge 1$ and $r\ge 0$ are nonnegative ...
  • 537
5 votes
1 answer
197 views

Relating $\int_0^1\frac{(\ln x)^{n-1}(\ln(1-z\,x))^p}{x}dx$ and $\int_0^1\frac{(\ln x)^{n}(\ln(1-z\,x))^{p-1}}{1-z\,x}dx$

This post, after a complicated analysis, evaluates the integral $$I=\int_0^1\frac{\ln^2(x)\,\ln^3(1+x)}xdx$$ simply as $$I =-\frac{\pi^6}{252}-18\zeta(\bar{5},1)+3\zeta^2(3)\tag1$$ where, $$\...
2 votes
1 answer
126 views

Higher order derivatives of the binomial factor

Let $p$,$l$ be positive integers and $\theta$ be a parameter. The question is to compute the following quantity: \begin{equation} \kappa^{(p)}_l := \left. \frac{\partial^p}{\partial \theta^p} \binom{\...
  • 9,303
3 votes
1 answer
65 views

Mixed partials of the Beta function B$(a,b)$ at $(1,0^+)$

In this post M.N.C.E gave the equality below $$\frac{\partial ^{5}}{\partial a^{3}\partial b^{2}}\mathrm{B}\left ( 1,0^{+} \right )=\left [ \frac{1}{b}+O\left ( 1 \right ) \right ]\left [ \left ( 12\...

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