What's New
About the Project
NIST
28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.4 Fourier Series

Contents

§28.4(i) Definitions

The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the z-plane. For n=0,1,2,3,,

28.4.1 ce2n(z,q) =m=0A2m2n(q)cos2mz,
28.4.2 ce2n+1(z,q) =m=0A2m+12n+1(q)cos(2m+1)z,
28.4.3 se2n+1(z,q) =m=0B2m+12n+1(q)sin(2m+1)z,
28.4.4 se2n+2(z,q) =m=0B2m+22n+2(q)sin(2m+2)z.

§28.4(ii) Recurrence Relations

28.4.5 aA0-qA2 =0,
(a-4)A2-q(2A0+A4) =0,
(a-4m2)A2m-q(A2m-2+A2m+2) =0,
m=2,3,4,, a=a2n(q), A2m=A2m2n(q).
28.4.6 (a-1-q)A1-qA3 =0,
(a-(2m+1)2)A2m+1-q(A2m-1+A2m+3) =0,
m=1,2,3,, a=a2n+1(q), A2m+1=A2m+12n+1(q).
28.4.7 (a-1+q)B1-qB3 =0,
(a-(2m+1)2)B2m+1-q(B2m-1+B2m+3) =0,
m=1,2,3,, a=b2n+1(q), B2m+1=B2m+12n+1(q).
28.4.8 (a-4)B2-qB4 =0,
(a-4m2)B2m-q(B2m-2+B2m+2) =0,
m=2,3,4,, a=b2n+2(q), B2m+2=B2m+22n+2(q).

§28.4(iii) Normalization

28.4.9 2(A02n(q))2+m=1(A2m2n(q))2=1,
28.4.10 m=0(A2m+12n+1(q))2 =1,
28.4.11 m=0(B2m+12n+1(q))2 =1,
28.4.12 m=0(B2m+22n+2(q))2 =1.

Ambiguities in sign are resolved by (28.4.13)–(28.4.16) when q=0, and by continuity for the other values of q.

§28.4(iv) Case q=0

28.4.13 A00(0) =1/2, A2n2n(0)
=1,
n>0,
A2m2n(0) =0,
nm,
28.4.14 A2n+12n+1(0) =1,
A2m+12n+1(0) =0,
nm,
28.4.15 B2n+12n+1(0) =1,
B2m+12n+1(0) =0,
nm,
28.4.16 B2n+22n+2(0) =1,
B2m+22n+2(0) =0,
nm.

§28.4(v) Change of Sign of q

28.4.17 A2m2n(-q) =(-1)n-mA2m2n(q),
28.4.18 B2m+22n+2(-q) =(-1)n-mB2m+22n+2(q),
28.4.19 A2m+12n+1(-q) =(-1)n-mB2m+12n+1(q),
28.4.20 B2m+12n+1(-q) =(-1)n-mA2m+12n+1(q).

§28.4(vi) Behavior for Small q

For fixed s=1,2,3, and fixed m=1,2,3,,

28.4.21 A2s0(q)=((-1)s2(s!)2(q4)s+O(qs+2))A00(q),
28.4.22 Am+2sm(q)Bm+2sm(q)}=((-1)sm!s!(m+s)!(q4)s+O(qs+1)){Amm(q),Bmm(q),
28.4.23 Am-2sm(q)Bm-2sm(q)}=((m-s-1)!s!(m-1)!(q4)s+O(qs+1)){Amm(q),Bmm(q).

For further terms and expansions see Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33).

§28.4(vii) Asymptotic Forms for Large m

As m, with fixed q (0) and fixed n,

28.4.24 A2m2n(q)A02n(q) =(-1)m(m!)2(q4)mπ(1+O(m-1))wII(12π;a2n(q),q),
28.4.25 A2m+12n+1(q)A12n+1(q) =(-1)m+1((12)m+1)2(q4)m+12(1+O(m-1))wII(12π;a2n+1(q),q),
28.4.26 B2m+12n+1(q)B12n+1(q) =(-1)m((12)m+1)2(q4)m+12(1+O(m-1))wI(12π;b2n+1(q),q),
28.4.27 B2m2n+2(q)B22n+2(q) =(-1)m(m!)2(q4)mqπ(1+O(m-1))wI(12π;b2n+2(q),q).

For the basic solutions wI and wII see §28.2(ii).