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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.4 Fourier Series

Contents
  1. §28.4(i) Definitions
  2. §28.4(ii) Recurrence Relations
  3. §28.4(iii) Normalization
  4. §28.4(iv) Case q=0
  5. §28.4(v) Change of Sign of q
  6. §28.4(vi) Behavior for Small q
  7. §28.4(vii) Asymptotic Forms for Large m

§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 aA0qA2 =0,
(a4)A2q(2A0+A4) =0,
(a4m2)A2mq(A2m2+A2m+2) =0,
m=2,3,4,, a=a2n(q), A2m=A2m2n(q).
28.4.6 (a1q)A1qA3 =0,
(a(2m+1)2)A2m+1q(A2m1+A2m+3) =0,
m=1,2,3,, a=a2n+1(q), A2m+1=A2m+12n+1(q).
28.4.7 (a1+q)B1qB3 =0,
(a(2m+1)2)B2m+1q(B2m1+B2m+3) =0,
m=1,2,3,, a=b2n+1(q), B2m+1=B2m+12n+1(q).
28.4.8 (a4)B2qB4 =0,
(a4m2)B2mq(B2m2+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)nmA2m2n(q),
28.4.18 B2m+22n+2(q) =(1)nmB2m+22n+2(q),
28.4.19 A2m+12n+1(q) =(1)nmB2m+12n+1(q),
28.4.20 B2m+12n+1(q) =(1)nmA2m+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 Am2sm(q)Bm2sm(q)}=((ms1)!s!(m1)!(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(m1))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(m1))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(m1))wI(12π;b2n+1(q),q),
28.4.27 B2m2n+2(q)B22n+2(q) =(1)m(m!)2(q4)mqπ(1+O(m1))wI(12π;b2n+2(q),q).

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