e517b11c-5a1f-4093-9253-02e0d5a829a5 24 4 5 decimal

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

x ProductLog e 1 exp : u x e > x ln x ln ln - x 1 e / + sqrt 1 e / sqrt - if 11 line : w 2 u * : u w - abs u abs / 1010 -^> w u : u u 2^x e u -^* + 1 u + / eval : 21 line while 31 line u 31 line :

x Lambert x 0.36787 - ≥ w w exp * x - 0 ≡ w 0 roots NO real roots -> granular if :

0.36787 - Lambert 0.992852729821595 -

0.36787 - ProductLog 0.992852729821989 -

Tip to plot

x W x ProductLog eval :

x ProductLog 15 decimals ≡

Plot eval(,,) Lambert W(x)

x W

Lambert [root conventional]

x Lambert

Example 1: solving a function

A general note: In mathematics [Advanced Algebra], solving for a function [finding its inverse] is rarely productive due to: 1. Non algebraic relationship. 2. Multiple solutions within the granularity of the binary computing machinery attempting various "numerical solvers". Maple 11 could always find a "RootOf" to be reconstructed within the responding or NOT responding "root". The Smath/Maple companion is more willing than Maple 11.

x α μ D μ x * α x * - exp * :

y x α μ D ≡ x solve maple α y * μ / - LambertW α / -

α 0.5 :

μ 1 :

An "InverseFunction" is a solving function. In this example, we constrain D(x,α,μ) to preserve the meaning of the model, but the φ(x) is not constrained as it may have some meaning in other applications. -------- Interpreting the example-------- "Given 'y' on D(x)=>" is sampling the alocotest in time. From the collected samples you can reconstruct "a personal model" from the theoretical "metabolisation model".

x φ α x * μ / - W α / - (:

x D x α μ D :

Given 'y' on D(x)=> ...

y 0.5 :

... read 'x' on φ(y) <=

x y φ :

sanity check

x 0.7148

x D 0.5

Alcohol metabolisation in blood

x 0 ≥ x α μ D useless for model if

The InverseFunction

x φ 1 1 sys

Example 2: solving a DE

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

t C Km t Vm Km / * (- C0 C0 Km / exp * ln Km / + exp Ω * :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

L 0 :

H 10 :

N 200 :

U L H N xd :

DE i 1 N 1 + range p i el U i el Vm Km / * (- C0 C0 Km / exp * ln Km / + exp : d i el Km p i el W * : 21 line for d 21 line :

DEsol U DE augment :

Title 2 1.85 DE solution in Pharmacology 10 black 15 mat :

DEsol Title 2 1 sys

t C t diff Vm t C * - Km t C + / ≡

jmGiraud: 2015/10/14

Example 3: Two more DE Lambert solution [from source]

x x diff x y (2^- x y (3^+ 0 ≡

0 y 0.2 ≡

t y t diff a t 1 - y * ≡

x Dsol 1144 x - exp * W + / :

a t y e a W t *^:

a 0.5 :

a 0 > real, positive ≡

x init 0.2 :

x 0 ≥ x Dsol undefined if x init 2 1 sys x 0 ≥ a x y undefined if

Example 4: Lambert solving algebraic expressions

n # :

u expr u n^e u^- :

u expr u solve maple n 1 n / - LambertW * -

1 n / - 1 e / - >

u expr u 2.5^e u^- :

These one will expand symbolic, it needs the Lambert complex branches, NOT given in this Smath document.

u expr u solve maple

u expr u 3^e u^- :

u expr u solve maple 313 / - LambertW * - 31 - 13 / - LambertW * - 31 i 3 sqrt * - 6 / LambertW * - 31 i 3 sqrt * + 6 / LambertW * - 41 mat

Proceed in parts

1 i 3 sqrt * - 6 / 0.1667 0.2887 i * -

16 / W i 0.2887 W (* - 0.1443 0.2295 i * -

1 - 13 / - LambertW unknown ≡

aa b v * + c z v -^* ≡ v solve maple z ln aa * - b z ln c * z ln aa * b / exp * b / LambertW * + z ln b * /

a 1 :

b 2 :

c π :

a b v * + c z v -^* ≡ v solve maple 2 π z ln * z sqrt * 2 / LambertW * z ln - 2 z ln * /

a b v * + c 10 v -^* ≡ v solve maple 2 π 10 sqrt * 10 ln * 2 / LambertW * - 10 ln + 210 ln * / -

a b v * + c e v -^* ≡ v solve maple 1 - 2 π 12 / exp * 2 / LambertW * + 2 /

2 xx^xx 2^≡ xx solve maple 2422 ln 2 / LambertW * 2 ln / - 31 mat

24 0.7667 - 31 mat

22 ln 2 / W * 2 ln / - 0.7667 -

aa xx^bb xx * + cc + 0 ≡ xx solve maple aa ln cc aa ln * bb / - exp * bb / LambertW bb * cc aa ln * + aa ln bb * / -

1 u e u^* + yy ≡ u solve maple 1 - yy + LambertW

u u e u^* + u u e u^* - / yy ≡ u solve maple 1 - yy + 1 yy + / ln

1 u e u^* + 1 u e u^* - / yy ≡ u solve maple 1 - yy + 1 yy + / LambertW

c 2 sqrt e π 31 mat :

a 2 :

b 0.019 :

i 1 c rows range sol i el a b v * + c i el 10 v^* ≡ v solve maple : for

A damned idiotic expansion

aa xx^bb xx * + cc + 0 ≡ xx solve maple aa ln cc aa ln * bb / - exp * bb / LambertW bb * cc aa ln * + aa ln bb * / -

2 xx^5 xx * - 0 ≡ xx solve maple 2 ln 5 / - LambertW 2 ln / - 1 - 2 ln 5 / - LambertW 2 ln / - 21 mat

1 - 2 ln 5 / - LambertW 2 ln / - unknown format ≡

Relevant calculus facts

xx LambertW xx diff maple xx LambertW 1 xx LambertW + (xx * /

xx LambertW xx int maple xx 11 - xx LambertW + (xx LambertW * + (* xx LambertW /