Greetings from the sunny Greece!
Let's see what Mathematica and Mind can do!
(I think this the deep scope of Vladimir's challenges!)
Let's define InverseJacobiAM (I think is not defined
as a built in function in Mathematica but this is not
a problem at all!)
$VersionNumber
5.2
InvJacAM[o_, k_] := Integrate[1/Sqrt[1 - k^2*Sin[u]^2], {u, 0, o},
GenerateConditions -> False]
So for the particular case we have
InvJacAM[I*ArcSinh[Sqrt[2]*z], I/Sqrt[3]]
EllipticF[I*ArcSinh[Sqrt[2]*z], -(1/3)]
Good!
Let's see the limit.
Limit[InvJacAM[I*ArcSinh[Sqrt[2]*z], I/Sqrt[3]], z -> Infinity]
Limit[EllipticF[I*ArcSinh[Sqrt[2]*z], -(1/3)], z -> Infinity]
It returns unevaluated. Bad!
I think the main problem is the expression ArcSinh[Sqrt[2]*z]
inside the EllipticF. For, as z->infinity, it diverges
logarithmically!
Indeed
(Simplify[#1, z > 0] & )[Normal[Series[ArcSinh[Sqrt[2]*z], {z,
Infinity, 4}]]]
-(3/(128*z^4)) + 1/(8*z^2) + Log[8]/2 + Log[z]
Limit[ArcSinh[Sqrt[2]*z], z -> Infinity]
Infinity
Let's investigate a bit the behavior of the function
EllipticF[I*ArcSinh[Sqrt[2]*z], -(1/3)],
graphically
Plot[Re[EllipticF[I*ArcSinh[Sqrt[2]*z], -(1/3)]], {z, 0, 30}, Axes -
False, Frame -> True]
Plot[Im[EllipticF[I*ArcSinh[Sqrt[2]*z], -(1/3)]], {z, 0, 30},
PlotRange -> {1.8, 1.9}, Axes -> False, Frame -> True]
It is apparent that for big z, the function tends to a constant
(complex) value.
Resorting to the integral representation of F[o,m] allows us to
calculate the value of
F[I infinity,m]
PowerExpand[Integrate[1/(Sqrt[1 - t^2]*Sqrt[1 - m*t^2]), {t, 0,
I*Infinity}, GenerateConditions -> False]]
I*EllipticK[1 - m]
Here m=-1/3. So, the requesting limit is (at last!)
I*EllipticK[1 - m] /. m -> -3^(-1)
I*EllipticK[4/3]
N[%]
1.4599026317063393 + 1.8675973343933385*I
Cheers
Dimitris
/ Vladimir Bondarenko :
Hello computer algebra adorers,
Is there a soul who can show in terms of CAS commands
how to calculate this limit
limit(InverseJacobiAM(I*arcsinh(sqrt(2)*z), I/sqrt(3)),
z= infinity);
(where InverseJacobiAM stands for the inverse of the
Jacobi amplitude function as it is defined in Maple) ?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
