Let-W-the-lambert-function-defined-as-W-xe-x-x-x-0-Prove-that-0-1-W-ulnu-u-du-2-2-

Question Number 80863 by ~blr237~ last updated on 07/Feb/20

L e t W t h e l a m b e r t f u n c t i o n d e f i n e d a s W ({x e}^{x}) = x x ⩾ 0

P r o v e t h a t \int_{0}^{1} \frac{W (- u l n u)}{u} d u = \frac{ζ (2)}{2}

Answered by Kamel Kamel last updated on 08/Feb/20

w (x) = - \underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n} n^{n - 1}}{n!} x^{n}

∴ Ω = \int_{0}^{1} w (- uLn (u)) \frac{du}{u} = - \underset{n = 1}{\sum^{+ \infty}} \frac{n^{n - 1}}{n!} \int_{0}^{1} u^{n - 1} {Ln}^{n} (u) du

\overset{u = e^{- t}}{=} - \underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n} n^{n - 1}}{n!} \int_{0}^{+ \infty} t^{n} e^{- nt} dt

= - \underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n}}{n^{2} n!} \int_{0}^{+ \infty} z^{n} e^{- z} dz = - \underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n}}{n^{2}}

= - (\frac{1}{4} ζ (2) - (ζ (2) - \frac{1}{4} ζ (2)) = \frac{ζ (2)}{2} = \frac{π^{2}}{12}