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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.22 Connection Formulas

  1. §28.22(i) Integer ν
  2. §28.22(ii) Noninteger ν

§28.22(i) Integer ν

28.22.1 Mcm(1)(z,h) =2π1ge,m(h)cem(0,h2)Cem(z,h2),
28.22.2 Msm(1)(z,h) =2π1go,m(h)sem(0,h2)Sem(z,h2),
28.22.3 Mcm(2)(z,h) =2π1ge,m(h)cem(0,h2)(fe,m(h)Cem(z,h2)+2πCm(h2)Fem(z,h2)),
28.22.4 Msm(2)(z,h) =2π1go,m(h)sem(0,h2)(fo,m(h)Sem(z,h2)2πSm(h2)Gem(z,h2)).

The joining factors in the above formulas are given by

28.22.5 ge,2m(h) =(1)m2πce2m(12π,h2)A02m(h2),
28.22.6 ge,2m+1(h) =(1)m+12πce2m+1(12π,h2)hA12m+1(h2),
28.22.7 go,2m+1(h) =(1)m2πse2m+1(12π,h2)hB12m+1(h2),
28.22.8 go,2m+2(h) =(1)m+12πse2m+2(12π,h2)h2B22m+2(h2),
28.22.9 fe,m(h) =π/2ge,m(h)Mcm(2)(0,h),
28.22.10 fo,m(h) =π/2go,m(h)Msm(2)(0,h),

where Anm(h2), Bnm(h2) are as in §28.4(i), and Cm(h2), Sm(h2) are as in §28.5(i). Furthermore,

28.22.11 Mcm(2)(0,h) =2/πge,m(h),
Msm(2)(0,h) =2/πgo,m(h),
28.22.12 fem(0,h2) =12πCm(h2)(ge,m(h))2cem(0,h2),
gem(0,h2) =12πSm(h2)(go,m(h))2sem(0,h2).

§28.22(ii) Noninteger ν

28.22.13 Mν(1)(z,h)=Mν(1)(0,h)meν(0,h2)Meν(z,h2).

Here meν(0,h2) (0) is given by (28.14.1) with z=0, and Mν(1)(0,h) is given by (28.24.1) with j=1, z=0, and n chosen so that |c2nν(h2)|=max(|c2ν(h2)|), where the maximum is taken over all integers .

28.22.14 Mν(2)(z,h)=cot(νπ)Mν(1)(z,h)1sin(νπ)Mν(1)(z,h).

See also (28.20.13) and (28.20.14).