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

§28.8 Asymptotic Expansions for Large q

  1. §28.8(i) Eigenvalues
  2. §28.8(ii) Sips’ Expansions
  3. §28.8(iii) Goldstein’s Expansions
  4. §28.8(iv) Uniform Approximations

§28.8(i) Eigenvalues

Denote h=q and s=2m+1. Then as h+ with m=0,1,2,,

28.8.1 am(h2)bm+1(h2)}2h2+2sh18(s2+1)127h(s3+3s)1212h2(5s4+34s2+9)1217h3(33s5+410s3+405s)1220h4(63s6+1260s4+2943s2+486)1225h5(527s7+15617s5+69001s3+41607s)+.

For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). For error estimates see Kurz (1979), and for graphical interpretation see Figure 28.2.1. Also,

28.8.2 bm+1(h2)am(h2)=24m+5m!(2π)1/2hm+(3/2)e4h(16m2+14m+732h+O(1h2)).

§28.8(ii) Sips’ Expansions

Let x=12π+λh1/4, where λ is a real constant such that |λ|<21/4. Also let ξ=2hcosx and Dm(ξ)=eξ2/4𝐻𝑒m(ξ)18.3). Then as h+

28.8.3 cem(x,h2) =C^m(Um(ξ)+Vm(ξ)),
sem+1(x,h2)sinx =S^m(Um(ξ)Vm(ξ)),


28.8.4 Um(ξ) Dm(ξ)126h(Dm+4(ξ)4!(m4)Dm4(ξ))+1213h2(Dm+8(ξ)25(m+2)Dm+4(ξ)+4! 25(m1)(m4)Dm4(ξ)+8!(m8)Dm8(ξ))+,
28.8.5 Vm(ξ) 124h(Dm+2(ξ)m(m1)Dm2(ξ))+1210h2(Dm+6(ξ)+(m225m36)Dm+2(ξ)m(m1)(m2+27m10)Dm2(ξ)6!(m6)Dm6(ξ))+,


28.8.6 C^m (πh2(m!)2)1/4(1+2m+18h+m4+2m3+263m2+262m+1082048h2+)1/2,
28.8.7 S^m (πh2(m!)2)1/4(12m+18h+m4+2m3121m2122m842048h2+)1/2.

These results are derived formally in Sips (1949, 1959, 1965). See also Meixner and Schäfke (1954, §2.84). For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5).

§28.8(iii) Goldstein’s Expansions

Let x=12πμh1/4, where μ is a constant such that μ1, and s=2m+1. Then as h+

28.8.8 cem(x,h2)cem(0,h2) =2m(1/2)σm(Wm+(x)(Pm(x)Qm(x))+Wm(x)(Pm(x)+Qm(x))),
sem+1(x,h2)sem+1(0,h2) =2m(1/2)τm+1(Wm+(x)(Pm(x)Qm(x))Wm(x)(Pm(x)+Qm(x))),


28.8.9 Wm±(x)=e±2hsinx(cosx)m+1{(cos(12x+14π))2m+1,(sin(12x+14π))2m+1,


28.8.10 σm 1+s23h+4s2+327h2+19s3+59s211h3+,
τm+1 2h14s2s2+326h7s3+47s210h2,
28.8.11 Pm(x) 1+s23hcos2x+1h2(s4+86s2+105211cos4xs4+22s2+57211cos2x)+,
28.8.12 Qm(x) sinxcos2x(125h(s2+3)+129h2(s3+3s+4s3+44scos2x))+.

§28.8(iv) Uniform Approximations

Barrett’s Expansions

Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). The approximations apply when the parameters a and q are real and large, and are uniform with respect to various regions in the z-plane. The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included.

Dunster’s Approximations

Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). These approximations apply when q and a are real and q. They are uniform with respect to a when 2qa(2δ)q, where δ is an arbitrary constant such that 0<δ<4, and also with respect to z in the semi-infinite strip given by 0zπ and z0.

The approximations are expressed in terms of Whittaker functions Wκ,μ(z) and Mκ,μ(z) with μ=14; compare §2.8(vi). They are derived by rigorous analysis and accompanied by strict and realistic error bounds. With additional restrictions on z, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii).

Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions meν(z,q)28.12(ii)) and modified Mathieu functions Mν(j)(z,h)28.20(iii)).

For related results see Langer (1934) and Sharples (1967, 1971).