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http://mathworld.wolfram.com/LambertW-Function.html http://www.jfepperson.org/2edition-web/lambert.pdf

Lambert W0, W1 principal branches

Lambert two principal branches x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 line

x ProductLog 15 decimals 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 u 51 line :

1 e / - 1010 -^+ W0 1 e / - ProductLog 21 mat 0.999976684 - 0.999999979 - 21 mat

e W0 e ProductLog 21 mat 1.000000000000000 1.000000000000000 21 mat

1 W0 1 ProductLog 21 mat 0.567143290409783 0.567143290409783 21 mat

recast ProductLog x W x ProductLog :

is approximalety 0.5671432904097838729999686622 [Neil Sloane] Ω 1 W ≡

Plot eval(,,) Lambert W(x)

x ProductLog

Lambert [root conventional]

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat 3 1 sys

Example: solving a DE

L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx - (dx i * + : for U 41 line :

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

Original solution from Mathematica t C Km t Vm Km / * (- C0 C0 Km / exp * ln Km / + exp ProductLog * :

L 0 : H 10 : N 200 : 13 mat U L H N xd : 21 line

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

DEsol U DE augment :

Title 1.5 1.85 DE solution in Pharmacology 12 red 15 mat :

DEsol Title 2 1 sys

t C t diff Vm t C * - Km t C + / ≡

Two DE solved in term of W [from source]

t y t diff a t 1 - y * ≡

x x diff x y (2^- x y (3^+ 0 ≡

0 y 0.2 ≡

a t y e a W t *^:

x Dsol 1144 x - exp * W + / :

a 0.25 :

a 0 > real, positive ≡

x init 0.2 :

a x y x 0 ≥ cases x Dsol x 0 ≥ cases x init 2 1 sys

Relevant calculus facts

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

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

This section is stand alone. Copy/paste in new WS project.

Lambert two principal branches x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 line

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 :

X x α μ D x diff x roots (: 2

Y X α μ D : 0.7358

recast LambertW  solutions y α μ Φ_0 α y * μ / - W0 α / - : y α μ Φ_1 α y * μ / - W1 α / - : 21 line

Explain:

The reply from Maple is correct, it does not have to educate

more about the user end use ... just like this. The complete

solutions are over the two Lambert branches [W0, W1] ...

implemented in this example, in full auto-mode.

]]> x α μ D 0 x ≤ ( x X ≤ ( & cases x α μ D X x ≤ ( x 12 ≤ ( & cases X 1 X 1 - 2 2 mat 3 1 sys

branch_0 solutions <= ... => branch_1 solutions

x α μ Φ_0 Y 2 Y 1 - 2 2 mat 2 1 sys x α μ Φ_1 Y 10 Y 10 - 2 2 mat 3 1 sys

Given 'y' on D(x,α,μ) ... y 0.25 :

... read 'x' on Φ_0(y,α,μ) <= x y α μ Φ_0 : 0.2888

... read 'x' on Φ_1(y,α,μ) <= x y α μ Φ_1 : 6.5234

sanity check x α μ D 0.25

sanity check x α μ D 0.25

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".

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 could always find a "RootOf" to be reconstructed within the responding or NOT responding "root". The Smath/Maple companion will help your advanced works.

Examples: LambertW solving algebraic expressions

1. Take the model of your choice => copy/paste in blank sheet,

2. Transport ProductLog(x) from above,

3. Work your example ... see image example below.

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Examples: Lambert solving algebraic expressions

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

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

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

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, complex branch ≡

Relevant calculus facts

x LambertW x diff maple #

x LambertW x int maple #