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## Re: [Help-gsl] Different value for mathieu_ce in Mathematica and GSL

 From: maxgacode Subject: Re: [Help-gsl] Different value for mathieu_ce in Mathematica and GSL Date: Sat, 18 Feb 2017 14:55:00 +0100 User-agent: Mozilla/5.0 (Windows NT 6.1; WOW64; rv:45.0) Gecko/20100101 Thunderbird/45.7.1

```Il 17/02/2017 23:17, Patrick Alken ha scritto:

```
```
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```N[MathieuC[MathieuCharacteristicA[0, -1], -1, 2*Pi/180]]
```
```1.41071
```

should be equivalent to
```
gsl_sf_mathieu_ce(0, -1.0, 2.0 * M_PI / 180.0)
```
which gives a totally different value: 0.99751942347886335.

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```

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Looking at Abramovitz and Stegun I found the following power serie for Ce(0,q,z) ( for small |q| ).
```
```
Ce(0,q,z) = ( 1/sqrt(2) ) * [ 1 - q * cos(2 z)/2 + q^2 * ((cos(4 z)/32) - 1/16) +........
```

for  q= -1 , z = 2 pi / 180

Ce(0,q,z) =~ 1.04 + ....

```
That is not proving anything but my guess is that GSL implementation agrees with Abramovitz and Stegun.
```
```
Moreover Scilab (using the Mathieu Toolbox from R.Coisson & G. Vernizzi, Parma University, 2001-2002.)
```
-->mathieu_ang_ce(0,-1, 2 * %pi / 180 ,1)
ans  =

0.9975194

again in agreement with GSL, Specfun and Abramovitz.

The Wolfram site says

```
"For nonzero q, the Mathieu functions are only periodic in z for certain values of a. Such characteristic values are given by the Wolfram Language functions MathieuCharacteristicA[r, q] and MathieuCharacteristicB[r, q] with r an integer or rational number. These values are often denoted a_r and b_r. In general, both a_r and b_r are multivalued functions with very complicated branch cut structures. Unfortunately,
```
there is no general agreement on how to define the branch cuts.

```
As a result, the Wolfram Language's implementation simply picks a convenient sheet. "
```

What are the values returned by

MathieuCharacteristicA[0, -1]

Hope this helps

Max

```