Elliptic integrals
11-02-2017, 05:42 PM (This post was last modified: 11-02-2017 05:54 PM by salvomic.)
Post: #2
 salvomic Senior Member Posts: 1,396 Joined: Jan 2015
RE: Elliptic integrals
These integrals are related to the Jacobi Elliptic Function (see in Wikipedia).

See here in the Library my version for the Prime.

About the two version of 1st kind integral I used in the two programs, they are related, being the same integral: x = sin(φ), then φ=ASIN(x), m=k^2...
The relation involve also sn(), cn(), dn() [sometimes called amplitude sine, amplitude cosine, delta amplitude

However I get values a bit different (see attachment): what kind of rounding is responsible of the difference, in this case?

A part of the examples in the attached image, another example: in the first case for the function ell1F() I test x=1 k=0.4 then ell1F(1,0.4) that returns 1.63999977306; in the second case (Jacobi) for the function Jacobi_fn() I test φ=ASIN(x), m=0.4^2 that returns 1.63999986587...

First case: the integral is int(1/(SQRT((1-t^2)*(1-(k^2)*t^2))),t,0,x);
Second case: the integral is int(1/(SQRT(1-m*SIN(θ)*SIN(θ))),θ,0,φ).
The *should* be the same value.
laTeX form (Wikipedia):
1st form
2nd form

Salvo

Attached File(s) Thumbnail(s)

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
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 Messages In This Thread Elliptic integrals - salvomic - 11-01-2017, 10:00 AM RE: Elliptic integrals - salvomic - 11-02-2017 05:42 PM RE: Elliptic integrals - AlexFekken - 11-03-2017, 01:54 PM RE: Elliptic integrals - salvomic - 11-03-2017, 03:41 PM RE: Elliptic integrals - Eddie W. Shore - 11-07-2017, 12:51 PM RE: Elliptic integrals - salvomic - 11-07-2017, 01:30 PM

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