Linked Questions

42 votes
7 answers
7k views

Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$

$\qquad\qquad$ Is there any closed form expression for the imaginary part of $~\text{Li}_3\bigg(\dfrac{1+i}2\bigg)$ ? Motivation: We already know that $~\Re\bigg[\text{Li}_3\bigg(\dfrac{1+i}2\bigg)\...
Lucian's user avatar
  • 48.5k
27 votes
6 answers
4k views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
Souvik's user avatar
  • 279
22 votes
3 answers
1k 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 ...
Anastasiya-Romanova 秀's user avatar
30 votes
2 answers
1k views

Evaluate $\int_0^{\pi/4}{(4\cot x\ln\sec x-x)\ln^2\tan xdx}$

Question: How can we prove $$I=\int_0^{\pi/4}{(4\cot x\ln\sec x-x)\ln^2\tan xdx}=\frac5{2304}\pi^4?$$ I confirmed, numerically, that it holds for 1000 decimal places. This integral came up when I was ...
Kemono Chen's user avatar
  • 8,679
25 votes
2 answers
1k views

How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

A while back I made a post asking for examples of integrals which evaluated to famous irrational constants (or constants that were very likely irrational but yet unproven to be). The top answer in ...
Robert Lee's user avatar
  • 7,273
9 votes
6 answers
882 views

Integral: $\int_{0}^{1}\frac{\arctan^{2}\left(x\right)}{x}dx$

Context: While working on a contour integral for fun, I stumbled upon the following integral: $$\int_{0}^{1}\frac{\arctan^{2}\left(x\right)}{x}dx.$$ I typed it into WolframAlpha and got that it equals ...
Accelerator's user avatar
  • 5,023
11 votes
4 answers
576 views

Prove that $\int_{0}^{\infty}\frac{(\arctan x)^3}{x^3}dx=\frac{3π}{2}\ln2-\frac{π^3}{16}$

Question: How to prove $$\int_{0}^{\infty}\frac{(\arctan x)^3}{x^3}dx=\frac{3π}{2}\ln2-\frac{π^3}{16}\: ?$$ I was able to prove $$\int_{0}^{\infty}\frac{(\arctan x)^2}{x^2}dx=π\ln2$$ using, $$f(x,y)=...
Paras's user avatar
  • 1,422
15 votes
2 answers
928 views

Compute polylog of order $3$ at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order $3$ at $\frac{1}{2}$ or $\operatorname{Li}_3\left(\frac{1}{2}\right)$, but how ...
Venus's user avatar
  • 11k
17 votes
3 answers
825 views

A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

In evaluating the integral given here it would seem that: $$\operatorname{Re} \operatorname{Li}_4 (1 + i) \stackrel{?}{=} -\frac{5}{16} \operatorname{Li}_4 \left (\frac{1}{2} \right ) + \frac{97}{...
omegadot's user avatar
  • 11.8k
16 votes
1 answer
1k views

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
user153012's user avatar
  • 12.4k
6 votes
3 answers
691 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\...
Ali Shadhar's user avatar
  • 25.8k
5 votes
1 answer
399 views

The identity $ \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \, \mathrm dx = 2 \sum_{n=0}^{\infty} \int_{a}^{b} p(x) \sin(anx) \, \mathrm dx$

Let $p(x)$ be a polynomial, and assume that $ \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \, \mathrm dx $ converges. How do you prove that $$ \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \...
Random Variable's user avatar
7 votes
2 answers
381 views

How to calculate $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}h_n}{n^3}$ where $h_n=\sum_{k=1}^{n}\frac{1}{2k-1}$?

Well known is the relationship $$\displaystyle\sum_{n=1}^{\infty}\frac{h_n}{n^3}=\operatorname{Li}_4(\frac{1}{2})+\frac{1}{24}\ln^42-\frac{\pi^2}{24}\ln^22+\frac{7}{8}\zeta(3)\ln2-\frac{53\pi^4}{5760}$...
user178256's user avatar
  • 5,517