Anecdotes please - quadratic equations in real life

[Albert Chan]

(12-30-2022 11:27 PM)Albert Chan Wrote:  e^W(a) = y = (y+a) / (ln(y)+1)

Let r = 1/e
If a = -r + h (tiny h), we have y = r + z (tiny z)

e^W(a) = (r+z) = (h+z) / ln(1+e*z)

Let H = e*h (known), Z = e*z (unknown), we have:

(1+Z) = (H+Z) / ln(1+Z)

H = (1+Z) * ln(1+Z) - Z = Z^2/2 - Z^3/6 + Z^4/12 - Z^5/20 + ...

2H ≈ Z^2 / (1+Z/3)
Z^2 - 2*(H/3)*Z - 2H ≈ 0
Z ≈ H/3 ± √((H/3)^2 + 2H) ≈ H/3 ± √(2H)

Divide both side by e, we have: y = r + z ≈ r + h/3 ± √(2*r*h)

e^W₀(z ≈ -1/e)   ≈   1/e + (z+1/e)/3 + sqrt ((2/e)*(z+1/e))

http://www.finetune.co.jp/~lyuka/technot...w-42s.html