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

[Albert Chan]

(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

Let ellipse perimeter = Pi (a+b) corr(h)
To simplify, assume a=1, we have h = (1-b)/(1+b) → b = (1-h)/(1+h)

Below Mathematica code tried to compare error of corr(h):

In[1]:= corr[h_] := 4*EllipticE[1-b^2] / (Pi(1+b)) /. b->(1-h)/(1+h)
In[2]:= ramj2[h_]:= 1 + 3h^2/(10 + (4-3h^2)^(1/2))
In[3]:= Series[corr[h] - ramj2[h], {h,0,12}] // Simplify
    → (3/131072) h^10 + (79/2097152) h^12 + O(h^14)

In[4]:= muir[b_,s_] := 2*((1+b^s)/2)^(1/s) / (1+b)
In[5]:= muir2[h_]:= muir[(1-h)/(1+h), 3/2+1/(32/h^2 - 3/5*h^2 - 4)]
In[6]:= Series[corr[h] - muir2[h], {h,0,12}] // Simplify
    → (1/737280) h^8 + (41/13762560) h^10 + (-31319/825753600) h^12 + O(h^14)

In[7]:= muir3[h_]:= muir[(1-h)/(1+h), 3/2+1/(32/h^2 - 217/360*h^2 - 4)]
In[8]:= Series[corr[h] - muir3[h], {h,0,12}] // Simplify
    → (181/61931520) h^10 + (-16717/440401920) h^12 + O(h^14)

New constant removed h^8 term.
Perimeter error is smaller until h > 0.3946 (eccentricity > 0.9009).