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

01142020, 11:47 PM
Post: #8




RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)
(01142020 05:23 PM)Gerson W. Barbosa Wrote: Here is still another approximation to the perimeter of the ellipse: 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 = (1b)/(1+b) → b = (1h)/(1+h) Below Mathematica code tried to compare error of corr(h): In[1]:= corr[h_] := 4*EllipticE[1b^2] / (Pi(1+b)) /. b>(1h)/(1+h) In[2]:= ramj2[h_]:= 1 + 3h^2/(10 + (43h^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[(1h)/(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[(1h)/(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). 

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