fa1941c4-5038-48d9-8f69-12970dee09f6 30 15 12 decimal

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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 dilog x k^k 2^/ k 1400 sum :

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

x polylog x p x q + :

Plolylog_1

x q x polylog x 1 ≥ cases 2 1 sys

Plolylog_1, complete plot

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

z Li0 z 1 z - (/ :

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

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

z Li1 1 z - ln - :

z Li_1 z 1 z - (2^/ :

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

x plot x Li0 x Li1 x Li_1 x Li_2 x Li_3 x Li_4 61 sys :

Polylogarithm series ... x < 1

x plot

1 - plot 0.5 - 0.6931 - 0.25 - 0 0.125 061 sys

Polylogarithm series ... x > 1, added PlolyLog_1

x plot x p x q + 2 1 sys x Li0 x Li1 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 + :

x digamma 1 - x ≤ cases

z digamma z 01 solve 0 digamma 21 mat 0.4861 0.5772 - 21 mat

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

Orphan PolyGamma functions

1 x Ψ 2 x Ψ 3 x Ψ 4 x Ψ 4 1 sys 0 x Ψ 1 x Ψ 2 1 sys

Venturi shape from "Orphan" Polygamma size 20 : shift 0.5 : X 2 : 13 mat n z Ψ 1 - (n 1 +^n! (* 1 z k + (shift + / (k 0 size sum * : x c 1 shift - x ≤ (x 2 ≤ (& cases : 31 line

Venturi nozzle of prescribed "size"

0 x Ψ x c * 1 x Ψ x c * X 5 X 5 - 2 2 mat 3 1 sys

00 Ψ 10 Ψ 21 sys 0 X Ψ 1 X Ψ 21 sys 21 mat 5.0081 - 5.0081 21 sys 2.4324 - 2.4324 21 sys 21 mat

Set integral accuracy 100 ε 1013 -^: x dilog t ln 1 t - / t 1 ε + x int : 21 line

1 ε + dilog 0.000000000000000

ε dilog 1.7209

x dilog x - dilog 2 1 sys

γ 0.5772156649015 :

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

x Gamma x diff x digamma x Gamma * ≡

x digamma 1 - x ≤ cases x digamma x Gamma * x digamma x Gamma 3 1 sys