85c2c2f3-af82-4001-8523-02b29ad03794 21 4 5 decimal

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Polylog_1 ...

... has an infinite series representation around a point 'a' close to the user point 'x' for end user that can't manage the "conjugate". You are not out of the bush if you can't manage Re(,)

a xx N Polylog_1infSeries 1 a - ln - xx a - (1 a - (/ + 1 - (n^n / xx a - (n^a 1 - (n 1 -^1 a - (* / * (n 2 N sum - Re :

x z1 1 x - ln :

x p 12 / 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 * - (* - :

x Polylog_1 x p x q + :

Plolylog_1, complete plot

x Polylog_1

a u u ε / + :

u 2.5 :

ε 100 :

a 2.525

a u u ε / - :

a 2.475

N 10 :

a u N Polylog_1infSeries 0.405465108108164 -

u Polylog_1 0.405465108108164 -

Polylog_1 roots [0, 2] ≡

x Δ x Polylog_1 x x ε / + x N Polylog_1infSeries - :

x x ε / + x N Polylog_1infSeries x Polylog_1 2 1 sys x Δ

Observe the dilogarithm

x dilog t ln 1 t - / t 1 x int ≡

zz 5 - 0 4.99 - range :

z dilog z Li2 ≡

datum determine the datum i 1 zz rows range Cumul i el x f x zz 1 el zz i el int - : 11 line for Cumul 31 line :

z Li2 z k^k 2^/ (k 138 sum :

x f 1 x - ln x / :

1 - x ≤ ( x 1 ≤ ( & x Li2 Undefined if

x f x z 1 el z i el int -

z 5 - 1 4.99 - range :

dilog Substract datum i 1 z rows range Cumul i el x f x z 1 el z i el int - : 11 line for z Cumul datum zz rows el - augment 31 line :

dilog cumulative integral

dilog 0 2 1 sys

The power series is limited ± 1. View the complete "dilog" from the cumulative integral shifted "datum". Range/mesh zz & z equally, datum is for zz ... 0.

The Mathcad 11/Maple manipulates the complex part x>1 in an unknown manner ... just demonstrative.

poly x 5 - 15 0.125 - (- range : i 1 x rows range u i el x i el : y i el 2 u i el polylog evalf maple : 21 line for u y augment 31 line :

n 1 :

x 1 ≥ complex ≡

n 9.9 - polylog evalf maple 23887627891000000000 / -

9.9 - Polylog_1 2.388762789 -

poly

Observe Maple does not plot finer than specified. As it looks, Smath is not authorised to collect a longer vector than 50. An error message is missing !

n 2 ≥ # polylog evalf

Black body polylog application

Black_body_Integrand u 3^e u -^* 1 e u -^- / u int ≡

An application of the polylog(n,x): the cumulative integral of the black body.

It has cumulative integral as function of polylog(n,x) the Maple solution agrees with Mathematica. Unfortunately that polylog solution is not exploitable in Smath. Note that the QuickPlot escapes the 0 singular integrand, the discrete form does NOT, thus ε in the discrete integrand.

μ 3^e μ -^* 1 e μ -^- / μ int maple μ μ μ 1 μ - exp - ln * 32 μ - exp polylog * - (* 63 μ - exp polylog * - (* 64 μ - exp polylog * -

Black body integrand

0 x ≤ x 3 ^ e x - ^ * 1 e x - ^ - / if

ε 1012 -^:

avoid singularity @ F(x=0)

U 010 0.02 range :

x F x 3^e x -^* 1 e x -^- (ε + / :

Integrand

i 1 U rows range f i el U i el F : for

black U f augment :

Black body cumulative integral Σ CumulativeIntegration i 1 U rows range Cumul i el x F x U 1 el U i el int : 11 line for U Cumul augment 31 line :

Black body complete system

black Σ 2 1 sys

Planck's law of spectral radiance

m sec 1 -^*

c 299792458 :

velocicity of light in vacuum

h 6.62606876 1034 -^* :

Planck's constant

joule sec *

k 1.3806503 1023 -^* :

joule K 1 -^*

Boltzmann's constant

λ T Γ 2 h * c 2^* λ 5^h c * k / 1 λ T * / * exp 1 - (* / :

2 π 5^* k 4^* 15 h 3^* c 2^* / T 4^* 5.6704 108 -^* ≡ 1 m 2^K 4^* / W *

T 1 :

Stefan's constant

λ f 2 π * c * k * T * λ 4^/ :

Be careful with the stuff in that section: Smath does not compute Stefan's constant !

Planck's law as function of λ(μ) , T(°K) ... for unit solid angle

x 10 6 - ^ * 500 Γ x 10 6 - ^ * 400 Γ x 10 6 - ^ * 300 Γ 3 1 sys