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Topic: Bug in Jacobian Amplitude?
Replies: 16   Last Post: Apr 13, 2013 1:35 PM

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clicliclic@freenet.de

Posts: 961
Registered: 4/26/08
Re: Bug in Jacobian Amplitude?
Posted: Apr 4, 2013 12:02 PM
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did schrieb:
>
> On Tuesday, 2 April 2013 19:48:53 UTC+2, clicl...@freenet.de wrote:

> >
> > The Mathematica and Maple answers are closely related: Re1 = pi - Re2,
> > Im1 = - Im2. This function has infinitely many branch points, and the
> > two systems appear to prefer different branches. However, I am having
> > trouble with the verification on Derive:
> >
> > ELLIPTIC_F(phi, m) := INT(1/SQRT(1 - m*SIN(t_*phi)^2), t_, 0, 1)
> >
> > ELLIPTIC_F(1.3306295147276587227 - 0.883132539714220814*#i, 3/4)
> >
> > 0.99660789047167089453 - 0.36927172197460334749*#i
> >
> > This integral doesn't look like 1 + 2*#i. What is wrong here?
> >
> > Along the same linear integration path from 0 to the Maple value, the
> > integrand passes though a branch cut of the square root, and I haven't
> > tried to work around this.
> >

>
> Of course I expected a problem with
> branch cut. From my Byrd & Friedman
> I tried the (supposed) two equivalent definitions:
>
> N[JacobiAmplitude[1+I*2, 3/4], 20]
> 1.3306295147276587227 - 0.8831325397142208140 I
>
> N[ArcTan[JacobiCN[1+I*2, 3/4] , JacobiSN[1+I*2, 3/4]], 20]
> 1.8109631388621345158 + 0.8831325397142208140 I
>
> and the Maple equivalent:
> evalf(JacobiAM(1+I*2,sqrt(3/4)),20);
> evalf(arctan(JacobiSN(1+I*2,sqrt(3/4)),JacobiCN(1+I*2,sqrt(3/4))),20);
> 1.8109631388621345158 + 0.88313253971422081406 I
> 1.8109631388621345158 + 0.88313253971422081409 I
>
> So Maple is consistent and matches MMA second
> expression. Why MMA should have a different
> definition of JacobiAmplitude? But, why it seems
> OK for other values? e.g.
> N[JacobiAmplitude[1 + I/2, 3/4], 20]
> 0.94644250288672303746 + 0.36465966602260927458 I
> N[ArcTan[JacobiCN[1 + I/2, 3/4], JacobiSN[1 + I/2, 3/4]], 20]
> 0.94644250288672303746 + 0.36465966602260927458 I
>


So Maple seems to use the full range -pi < Re(am) <= pi for the angle
am, in accordance with am(u, k) = arctan(sn(u, k), cn(u, k)), whereas
Mathematica seems to map the angle am into the range pi/2 < Re(am) <=
pi/2, in accordance with am(u, m) = arctan(sn(u, m)/cn(u, m)).

Meanwhile I have succeeeded to verify two of your am values :) - a
prefactor phi had been missing in front of my integral:

ELLIPTIC_F(phi, m) := phi*INT(1/SQRT(1 - m*SIN(t_*phi)^2), t_, 0, 1)

ELLIPTIC_F(1.3306295147276587227 - 0.883132539714220814*#i, 3/4)

1 - 1.3715007096251920858*#i

ELLIPTIC_F(0.94644250288672303746 + 0.36465966602260927458*#i, 3/4)

1 + 0.5*#i

But I am still having trouble with the imaginary part in the first
example! :(

Martin.



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