170
14 Mathieu Functions
and
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1 − q q 0 0 · · ·
q 9 q 0 · · ·
0 q 25 q · · ·
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
q (2r − 1) 2 ) q
q
. . .
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B
(2n+1)
1
B
(2n+1)
3
B
(2n+1)
5
. . .
. . .
B
(2n+1)
2r+1
. . .
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= b 2k+1
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B
(2n+11)
1
B
(2n+1)
3
B
(2n+1)
5
. . .
. . .
B
(2n+1)
2r+1
. . .
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,
(14.17b)
where the eigenvalues a k start with k = 0 and b k with k = 1, Fig. 14.1.
Therefore, with the MATLAB function eigs, we can quickly and easily compute
the eigensolutions and, from the Taylor expansions, the corresponding Mathieu
function.
Basic Solutions
The functions w I and w I I are the even (I) and odd (II) parity solutions of the
Mathieu equation (14.1). The initial conditions are
w I (z; a, q) = 1, and
d
dz
w I (z; a, q) = 0 and
(14.18)
w I I (z; a, q) = 0, and
d
dz
w I I (z; a, q) = 1.
(14.19)
For a equal to an eigenvalue a k , we get
w I (z; a k , q) · ce k (0, q) = ce k (z, q),
(14.20)
and if a equals an eigenvalue b l
w I I (z; b l , q) ·
d
dz
se l (0, q) = se l (z, q).
(14.21)
The evaluation of w I and w I I is based on a direct integration of the differential
equation with the MATLAB function ode113.
Second Solutions
As already mentioned, see Eq. (14.4), a second linear independent solution exists,
referred as f e n (z, q) for the functions ce n (z, q) and ge n (z, q) for the functions