fa1941c4-5038-48d9-8f69-12970dee09f6 11 4 5 decimal

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

Polylogarithm [Jonquière function]

x p 12 / 1 x - ln * - :

x z1 1 x - ln :

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

x p x q + x PolyLog_1 ≡

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

Plolylog_1

x q x 1 ≥ x p x q + Undefined if 2 1 sys

Plolylog_1, complete plot

x p x q +

z Li_2 z 1 z + (* 1 z - (3^/ :

z Li.1 1 z - ln - :

z Li_3 z 14 z * + z 2^+ (* 1 z - (4^/ :

z Li.0 z 1 z - (/ :

z Li_1 z 1 z - (2^/ :

z Li_4 z 1 z + (* 110 z * + z 2^+ (* 1 z - (5^/ :

x plot x Li.1 x Li.0 x Li_1 x Li_2 x Li_3 x Li_4 61 sys :

Polylogarithm series ... x < 1

x plot

Polylogarithm series ... x > 1, added PlolyLog_1

x plot x p x q + 2 1 sys

PolyGamma Functions

n z Ψ 1 - (n 1 +^n! (* 1 z k + (1 + / (k 010 sum * :

PolyGamma functions ψ(n,z)

1 x Ψ 2 x Ψ 3 x Ψ 3 1 sys

γ 0.5772156649015 :

z digamma γ - z n n z + (* / (n 120 sum + :

1 - x ≤ x digamma undefined if

DISAMBIGUATION: In applied/advanced maths, many functions are expressed in companionship to other functions and naturally within the range of the valid "solution". For digamma, we expose part of the "meaningful digamma". Note: digamma convergence depends upon the length of the power series Σ. In practical Engineering/Scientific works, you may have a data set that has underlying "digamma", for sure you could "fit weirdly", but why not fit "wisely" ?