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

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Waldek Hebisch

Posts: 227
Registered: 12/8/04
Re: Bug in Jacobian Amplitude?
Posted: Apr 4, 2013 9:00 PM
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did <didier.oslo@hotmail.com> wrote:
>
> 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


This definition is suboptimal, because it forces discontinuity
on real line. The definition I recall were continuous
(increasing) on real line.

In Abramowitz and Stegun the only definition I have found is
via equations JacobiSn(x, m) = sin(JacobiAmplitude(x, m))
and JacobiCn(x, m) = cos(JacobiAmplitude(x, m))

However Mathematica violates those equations:

In[1]:= N[JacobiSN[1+I*2, 3/4], 20]

Out[1]= 1.37533383624444180490 - 0.23845671888060964165 I

In[2]:= N[Sin[JacobiAmplitude[1+I*2, 3/4]], 20]

Out[2]= 1.37533383624444180490 - 0.23845671888060964165 I

In[3]:= N[JacobiCN[1+I*2, 3/4], 20]

Out[3]= -0.33681033299005352886 - 0.97371595177876063808 I

In[4]:= N[Cos[JacobiAmplitude[1+I*2, 3/4]], 20]

Out[4]= -0.33681033299005352886 - 0.97371595177876063808 I

In[5]:= N[JacobiAmplitude[1+I*2, 3/4], 20]

Out[5]= 1.3306295147276587227 - 0.8831325397142208140 I

In[6]:= N[Cos[1.3306295147276587227 - 0.8831325397142208140*I], 20]

Out[6]= 0.336810332990053529 + 0.973715951778760638 I

Note that in Out[4] Mathematica pretends that the equation
for JacobiCN holds, but numerial values in Out[5] and Out[6]
show that it is not satisfied. Since the sign of the last
value is incorrect and equation for JacobiSN seem to hold
one can guess that Mathematica simply uses
ArcSin[JacobiSN[x, m]] for computing JacobiAmplitude

>
> 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


This is consistent with the ArcSin definition.

--
Waldek Hebisch
hebisch@math.uni.wroc.pl



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