Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)

[Gerson W. Barbosa]

(01-14-2020 11:47 PM)Albert Chan Wrote:  

(01-14-2020 05:23 PM)Gerson W. Barbosa Wrote:  Here is still another approximation to the perimeter of the ellipse:

p ≈ 2π[(aˢ + bˢ)/2]¹ᐟˢ

where

s = 3/2 + 1/(32/h² - ⅗h² - 4)

and where

h = [(a - b)/(a + b)]

How did you get "⅗" from the keyboard ?
BTW, if you change "⅗" to "217/360", above formula is always better than Ramanujan II

a) I simply copied it from Wikipedia ;

b) Great! Thanks for your analysis, results and experimenting with Mathematica.

I’ve used a much more modest tool, the hp33s Solver:

XROOT(X:(A^X+B^X)÷)=C

By making, for instance, A=3, B=2, C=2.52506313496 (the equivalent radius) and solving for X, I got X = 1.50125631949. The inverse of that result after the subraction of 3/2 is 795.975887056, which divided by 25 (1/h)² gives 31.8390348302. Likewise the same procedure on a couple of other examples always would return answers close to 32. Once this constant is settled, the second constant, 4, is evident: 25×32 - 795.975887056 = 4.02412294. And so on...