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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.20 Definitions and Basic Properties


§28.20(i) Modified Mathieu’s Equation

When z is replaced by ±iz, (28.2.1) becomes the modified Mathieu’s equation:

28.20.1 w′′-(a-2qcosh(2z))w=0,

with its algebraic form

28.20.2 (ζ2-1)w′′+ζw+(4qζ2-2q-a)w=0,

§28.20(ii) Solutions Ceν, Seν, Meν, Fen, Gen

28.20.3 Ceν(z,q) =ceν(±iz,q),
28.20.4 Seν(z,q) =iseν(±iz,q),
28.20.5 Meν(z,q) =meν(-iz,q),
28.20.6 Fen(z,q) =ifen(±iz,q),
28.20.7 Gen(z,q) =gen(±iz,q),

§28.20(iii) Solutions Mν(j)

Assume first that ν is real, q is positive, and a=λν(q); see §28.12(i). Write

28.20.8 h=q(>0).

Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ζ1/2e±2ihζ as ζ in the respective sectors |ph(iζ)|32π-δ, δ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions Mν(3)(z,h), Mν(4)(z,h) such that

28.20.9 Mν(3)(z,h)=Hν(1)(2hcoshz)(1+O(sechz)),

as z+ with -π+δz2π-δ, and

28.20.10 Mν(4)(z,h)=Hν(2)(2hcoshz)(1+O(sechz)),

as z+ with -2π+δzπ-δ. See §10.2(ii) for the notation. In addition, there are unique solutions Mν(1)(z,h), Mν(2)(z,h) that are real when z is real and have the properties

28.20.11 Mν(1)(z,h)=Jν(2hcoshz)+e|(2hcoshz)|O((sechz)3/2),
28.20.12 Mν(2)(z,h)=Yν(2hcoshz)+e|(2hcoshz)|O((sechz)3/2),

as z+ with |z|π-δ.

For other values of z, h, and ν the functions Mν(j)(z,h), j=1,2,3,4, are determined by analytic continuation. Furthermore,

28.20.13 Mν(3)(z,h) =Mν(1)(z,h)+iMν(2)(z,h),
28.20.14 Mν(4)(z,h) =Mν(1)(z,h)-iMν(2)(z,h).

§28.20(iv) Radial Mathieu Functions Mcn(j), Msn(j)

For j=1,2,3,4,

28.20.15 Mcn(j)(z,h) =Mn(j)(z,h),
28.20.16 Msn(j)(z,h) =(-1)nM-n(j)(z,h),

§28.20(v) Solutions Ien, Ion, Ken, Kon

28.20.17 Ien(z,h) =i-nMcn(1)(z,ih),
28.20.18 Ion(z,h) =i-nMsn(1)(z,ih),
28.20.19 Ke2m(z,h) =(-1)m12πiMc2m(3)(z,ih),
Ke2m+1(z,h) =(-1)m+112πMc2m+1(3)(z,ih),
28.20.20 Ko2m(z,h) =(-1)m12πiMs2m(3)(z,ih),
Ko2m+1(z,h) =(-1)m+112πMs2m+1(3)(z,ih).

§28.20(vi) Wronskians

28.20.21 𝒲{Mν(1),Mν(2)} =-𝒲{Mν(2),Mν(3)}=-𝒲{Mν(2),Mν(4)}=2/π,
𝒲{Mν(1),Mν(3)} =-𝒲{Mν(1),Mν(4)}=-12𝒲{Mν(3),Mν(4)}=2i/π.

§28.20(vii) Shift of Variable

For n=0,1,2,,

28.20.23 Mc2n(j)(z±12πi,h) =Mc2n(j)(z,±ih),
Ms2n+1(j)(z±12πi,h) =Mc2n+1(j)(z,±ih),
28.20.24 Mc2n+1(j)(z±12πi,h) =Ms2n+1(j)(z,±ih),
Ms2n+2(j)(z±12πi,h) =Ms2n+2(j)(z,±ih).

For s,

28.20.25 Mν(1)(z+sπi,h) =eisπνMν(1)(z,h),
Mν(2)(z+sπi,h) =e-isπνMν(2)(z,h)+2icot(πν)sin(sπν)Mν(1)(z,h),
Mν(3)(z+sπi,h) =-sin((s-1)πν)sin(πν)Mν(3)(z,h)-e-iπνsin(sπν)sin(πν)Mν(4)(z,h),
Mν(4)(z+sπi,h) =eiπνsin(sπν)sin(πν)Mν(3)(z,h)+sin((s+1)πν)sin(πν)Mν(4)(z,h).

When ν is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).