14.2 Fundamental Equations and Computation
171
se n (z, q) [2, 3]. Thus,
f e n (z, q) = C n (q) (z ce n (z, q) + f n (z, q)), and
(14.22)
ge n (z, q) = S n (q) (z se n (z, q) + g n (z, q)).
(14.23)
Hence, for n ∈ N, f 2n (z, q) is π-periodic and f 2n+1 (z, q) is π-antiperiodic, both
with parity odd, and g 2n+1 (z, q) is π-antiperiodic and g 2n+2 (z, q) is π-periodic,
both g’s with parity even. The factors C n (q) and S n (q) are normalized such that
(C n (q))
2
2π
0
(f n (z, q))
2 dz = (S n (q))
2
2π
0
(g n (z, q))
2 dz = π .
(14.24)
For the computation of f e n (z, q)andge n (z, q), we integrate the differential
equation (14.1) directly using the MATLAB-function ode113.
For f e n (z, q), we start with the initial values (0, 1), which results in ˜
f e n (z, q) =
c · f e n (z, q), with c an unknown constant. Because f n (π, q) = 0, we could
compute an intermediate constant via ˜
c ˜
f e n (π, q) = π · ce n (π, q). This allows
us to compute ˜
f n (z, q) = ˜
c ˜
f e n (z, q) − z · ce n (z, q), and due to Eq. (14.24),
C n (q) 2 = |π/
2π
0 ( ˜
f n (z, q) 2 | and thus the scaling c = C n (q) · ˜
c.
For ge n (z, q), we start by integrating Eq. (14.1) with the initial value (1, 0),
which gives ˜
ge n (z, q) ∝ ge n (z, q). To compute the scaling factor c, we have to
evaluate the functions at two points (r, r + π), with r a random number between
]0, 0.5], because ge n (0, q) = 0. As g n (r, q) = (−1) n g n (r + π, q), we can
compute (similar to f e n ) the proportional constant ˜
c and, with the normalization
condition (14.24), the scaling factor between ge n and ˜
ge n .
Mathieu Functions of Non-integer Order
For non-integer ν, the series expansion of the Mathieu functions reads
me ν (z, q) =
∞
r=−∞
c
(ν)
2r (q) exp[i(ν + 2 r) z]
(14.25a)
ce ν (z, q) =
∞
r=−∞
c
(ν)
2r (q) cos[(ν + 2 r) z]
(14.25b)
se ν (z, q) =
∞
r=−∞
c
(ν)
2r+1 (q) sin[(ν + 2 r) z],
(14.25c)
and the coefficients satisfy
q c
(ν)
2r+2 −
a − (ν + 2r)
2
q c
(ν)
2r + q c
(ν)
2r−2 = 0,
(14.25d)