14.3 Programs
175
q: 1.2500
n: [1 3 5]
scale: []
info: {’Mathieu eigenfunction ce(z,q);
antiperiod pi and even parity’
Because n = 5, ce 1 (z, q), ce 3 (z, q), and ce 5 (z, q) will be evaluated. The first
row of “obj.value” are the function values of ce 1 (z, q), the second ce 3 (z, q), and so
on.
objCe = mathieuFun(’Cehyp’, 5, 1.25, z, ’eigen’,2);
>> objCe
objCe =
mathieuFun with properties:
value: [6x100 double]
z: [100x1 double]
eigen: {[3x3 double] [101x3 double]}
q: 1.2500
n: [1 3 5]
scale: []
info: {’modified Mathieu eigenfunction Ce(z,q) and
dCe(z,q); evaluation based on solving
differential equation’}
The evaluation of the modified Mathieu function is based on a direct integration
of the differential equation (14.2). Therefore, the function values and its first
derivative are computed. Here, the first row in “objCe.value” are the function values
of Ce 1 (z, q), the second row are the function values of Ce 3 (z, q), and so on, and the
fourth row are the function values of the first derivative
d
dz Ce 1 (z, q), followed by
d
dz Ce 3 (z, q) and
d
dz Ce 5 (z, q). “objCe.eigen” is a cell variable. The first cell element
are the eigenvalues in a diagonal matrix, and the second one are the eigenvectors.
Functions f e n (z, q), ge n (z, q) F e n (z, q), and Ge n (z, q)
The syntax is, e.g., obj=mathieuFun(’fe’, n, q, ‘zend’, 2 * pi).
The evaluation of these functions and their first derivative is based on integrating
the corresponding differential equation. For the Mathieu functions f e and ge, the
default integration interval runs from 0 to 2π and for the modified Mathieu functions
F e and Ge from 0 to 1. For the input argument “n,” only positive integers are
allowed. The following name–value pairs are supported: “eigen” and “scale.” For
the name-value pair (’scale’, 1) the scaling factor C n (q) or S n (q) will be uncovered
in the object property “scale”, and for the pair (‘scale’, 2), the scaling c is also
uncovered as second vector component. Example:
>> objfe = mathieuFun(’fe’, 5, 1.25, ’scale’, 2);
>> objfe
objfe =
mathieuFun with properties:
value: [377x2 double]
z: [377x1 double]
eigen: []
q: 1.2500