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[MUG] [Q]: Maple's EllipticF(z, k) vs Mathematica's
EllipticF[ArcSin[z], k^2]
| [MUG] [Q]: Maple's EllipticF(z, k) vs Mathematica's
EllipticF[ArcSin[z], k^2] |
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Author: Vladimir Bondarenko
Posted: Sun, 30 Jun 2002 14:28:23 +0400
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>> From: Vladimir Bondarenko "vvb"
Hello.
According to Maple Help
The incomplete elliptic integral EllipticF is defined by
EllipticF(z,k) = int(1/sqrt(1-t^2)/sqrt(1-k^2*t^2),t=0..z)
Translation into Mathematica yields
Integrate[1/Sqrt[1-t^2]/Sqrt[1-k^2*t^2],{t,0,z}]
EllipticF[ArcSin[z], k^2]
Therefore, we can make the following comparison.
Maple 8 Mathematica 4.2
EllipticF(z,k) EllipticF[ArcSin[z], k^2]
Let us substitute concrete values.
These outputs coinside:
evalf(EllipticF(1/2,1/2),20); N[EllipticF[ArcSin[1/2], 1/4], 20]
.52942862705190581774 0.52942862705190581774
Another perfect agreement:
evalf(EllipticF(I,1/2),20); N[EllipticF[ArcSin[I], 1/4], 20]
0.+.85122374907118540906*I 0.85122374907118540906 I
Yet another perfect agreement:
evalf(EllipticF(1,I),20); N[EllipticF[ArcSin[1], -1], 20]
1.3110287771460599052 1.3110287771460599052
However, these outputs differ
evalf(EllipticF(I,I),20); N[EllipticF[ArcSin[I], -1], 20]
0.+1.3085903338656260177*I 1.3110287771460599052 I
Question 1) Which of them is correct? (my idea is, the Mathematica's one)
Question 2) How to prove your answer to 1) in an easy and elegant way?
Naturally, one can use the AGM idea but this looks somewhat
tedious (me, lazy...)
Thanx for your help in advance!
Best,
Vladimir Bondarenko
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