nice-calculus-prove-that-0-1-ln-x-2-li-3-x-1-x-dx-2-3-6-

Question Number 123387 by mnjuly1970 last updated on 25/Nov/20

\dots n i c e c a l c u l u s \dots

p r o v e t h a t ::

Ω = \int_{0}^{1} \frac{{(l n (x))}^{2} {l i}_{3} (x)}{1 - x} d x

\overset{? ? ?}{=} ζ^{2} (3) - ζ (6) ✓

Commented by talminator2856791 last updated on 25/Nov/20

what is {li}_{3} ?

Commented by mnjuly1970 last updated on 25/Nov/20

{l i}_{3} (x) = \int_{0}^{x} \frac{{l i}_{2} (t)}{t} d t = \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{n^{3}}

t r i l o g a r i t h m f u n c t i o n .

{l i}_{z} (x) = \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{n^{z}}

e x a m p l e : {l i}_{2} (1) = ζ (2) = \frac{π^{2}}{6}

{l i}_{3} (1) = ζ (3), {l i}_{2} (- 1) = - η (2) = - \frac{π^{2}}{12}

, \dots .