10d8b8cd-d258-4075-bad4-031d0937769d 12 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Polylogarithm [Jonquière function]

x p 12 / 1 x - ln * - :

x z1 1 x - ln :

x q 12 / x z1 Re i x z1 Im * - (* - :

x z1 Re i x z1 Im * - conjugate of -> ≡ x z1 ≡

x polylog x p x q + :

specific values 1.5 polylog 3 polylog 2 ln 31 mat 0.693147180559945 0.693147180559945 - 0.693147180559945 31 mat

Plolylog elements

x q x polylog x 1 ≥ cases 2 1 sys

Plolylog(x), complete plot

x polylog 1 10 1 10 - 2 2 mat 2 1 sys

Jonquière x n Ji x k^k n^/ k 140 sum :

x c 11 - x ≤ (x 1 ≤ (& cases :

x Jonquiere x 1 Ji x c * x 2 Ji x c * x 3 Ji x c * x 7 Ji x c * 41 sys :

x Jonquiere x Jonquiere

x x x 1 x - exp - ln * 32 x - exp polylog * - (* 63 x - exp polylog * - (* 64 x - exp polylog * -

Substitute each polylog term by its "Jonquière" Σseries ... x JonQ x x x 1 x - exp - ln * 3 x - exp 2 Ji * - (* 6 x - exp 3 Ji * - (* 6 x - exp 4 Ji * - :

x JonQ

x f x JonQ :

h 0.1 :

Incremental  Derivative x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* (:

x g 0.5 :

x q x h f deriv :

x q x g - x 07 solve 0.9506 6.1897 21 mat

slow, incremental rescues project derivative operator fails JonQ(x)

x h f deriv