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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.26 Asymptotic Approximations for Large q

  1. §28.26(i) Goldstein’s Expansions
  2. §28.26(ii) Uniform Approximations

§28.26(i) Goldstein’s Expansions


28.26.1 Mcm(3)(z,h) =eiϕ(πhcoshz)1/2(Fcm(z,h)iGcm(z,h)),
28.26.2 iMsm+1(3)(z,h) =eiϕ(πhcoshz)1/2(Fsm(z,h)iGsm(z,h)),


28.26.3 ϕ=2hsinhz(m+12)arctan(sinhz).

Then as h+ with fixed z in z>0 and fixed s=2m+1,

28.26.4 Fcm(z,h)1+s8hcosh2z+1211h2(s4+86s2+105cosh4zs4+22s2+57cosh2z)+1214h3(s5+14s3+33scosh2z2s5+124s3+1122scosh4z+3s5+290s3+1627scosh6z)+,
28.26.5 Gcm(z,h)sinhzcosh2z(s2+325h+129h2(s3+3s+4s3+44scosh2z)+1214h3(5s4+34s2+9s647s4+667s2+283512cosh2z+s6+505s4+12139s2+1039512cosh4z))+.

The asymptotic expansions of Fsm(z,h) and Gsm(z,h) in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively.

For additional terms see Goldstein (1927).

§28.26(ii) Uniform Approximations

See §28.8(iv). For asymptotic approximations for Mν(3,4)(z,h) see also Naylor (1984, 1987, 1989).