       Re: Mathematica 6.0 bug in computing MathieuC

• To: mathgroup at smc.vnet.net
• Subject: [mg80403] Re: Mathematica 6.0 bug in computing MathieuC
• From: T.bakri at gmail.com
• Date: Tue, 21 Aug 2007 05:07:20 -0400 (EDT)
• References: <fabgsc\$ien\$1@smc.vnet.net>

```On Aug 20, 9:46 am, chuck009 <dmili... at comcast.com> wrote:
> > Dear all,
> > Something is still worng with the algorithm behind
> > computing MathieuC. It
> > reprts, for instance that MathieuC is pi-periodic
> > when its not.
>
> Hey T, I was unable to cause the following code to fail to give a pi-periodic function "poly" for any natural number r and parameter q:
>
> r = 4;
> q = -3 - 3*I
> a = N[MathieuCharacteristicA[r, q]]
> poly = MathieuC[a, q, x]/Exp[I*r*x];
> p1 = Plot[Im[poly], {x, 0, 10}];
> linelist = (Line[{{#1*Pi, -3}, {#1*Pi, 3}}] & ) /@ Range[1, 3]
> Show[{p1, Graphics[linelist]}]
>
> where the division by Exp[irx] is a consequence of Floquet's Theorem.

Take the parameter c such that MathieuCharacteristicB[2, c]=1.  One
can easilty numerically compute c. We find
c=-6.433074585014675206670244658478726486156264965498297244484.
Clearly, we have MathieuS(1,c,t) is Pi-periodic. However
MathieuC(1,c,t) should oscillate but is NOT Pi-periodic. Moreover the
Wronski-determinant ( i.e. MathieuS(1,c,t)*MathieuCPrime(1,c,t) -
MathieuC(1,c,t)*MathieuSPrime(1,c,t)) is in this case time-independent
and must be equal to -1. Mathematica 6.0 does this all worng. You can
check it yourself.

Regards,
T. Bakri

```

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