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I have been playing around with Mathematica's Mathieu functions as I am interested in the their limit when q tends to infinity. In particular I have been plotting several asymptotic forms, the most clearly presented of which I have found to be those outlined in the work of Frenkel and Portugal http://www.lncc.br/~portugal/Mathieu.pdf

r = 36; 
q = 5000;
Rasterize[ Plot[{Sqrt[1/\[Pi]]MathieuC[MathieuCharacteristicA[r, q],
 q, z/2 + \[Pi]/2]/Sign[MathieuC[MathieuCharacteristicA[r, q], q, 
\[Pi]/2]], (-1)^(r/2) Sqrt[q^(1/4)/(Sqrt[2 \[Pi]]
r!)] ParabolicCylinderD[r, 2 q^(1/4) Sin[z/2]] }, 
{z, -8 \[Pi]/q^(1/4), 8 \[Pi]/q^(1/4)}, PlotRange -> Full, 
PlotStyle -> {{Blue, Dashed}, {Orange, Dotted}}, 
PlotLegends -> "Expressions", ImageSize -> Large] 
Sign[MathieuC[MathieuCharacteristicA[r, q], q, \[Pi]/2]] Clear[r, q]

In particular I would like to point out equation (42) which I have included a comparison of Failure of the Mathieu function at large q for r=6 and q=10^9

Note that the qualitative features of the Mathieu equation are butchered. Namely the solution plotted has 4 nodes, rather than 6. This happens right around q=10^9 for some reason. At a lower value of q=10^8 you can see everything works perfectly.

Good agreement between asymptotics and Mathematica's "exact" Mathieu function

Ryan Plestid
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    You need to post the executable Mathematica code in code blocks so that it can be copied and pasted into a notebook. – Bob Hanlon Dec 31 '17 at 02:50
  • Please read the description of the [tag:bugs] tag. – Michael E2 Dec 31 '17 at 05:57
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    There are several problems with Mathematica's Mathieu functions reported here and on stackoverflow. – Michael E2 Dec 31 '17 at 06:03
  • @MichaelE2. In particular, it is likely that this is the same error as in this post. – march Dec 31 '17 at 21:37
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    Check individual values rather than a plot. I don't get the same plot as yours, but when I evaluate your expression I match the expected values. I assume the difference is because plotting must convert from exact precision, whereas your expression is properly using exact values. – Bill Watts Jan 01 '18 at 21:44

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