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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.20 Definitions and Basic Properties

  1. §28.20(i) Modified Mathieu’s Equation
  2. §28.20(ii) Solutions Ceν, Seν, Meν, Fen, Gen
  3. §28.20(iii) Solutions Mν(j)
  4. §28.20(iv) Radial Mathieu Functions Mcn(j), Msn(j)
  5. §28.20(v) Solutions Ien, Ion, Ken, Kon
  6. §28.20(vi) Wronskians
  7. §28.20(vii) Shift of Variable

§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′′(a2qcosh(2z))w=0,

with its algebraic form

28.20.2 (ζ21)w′′+ζw+(4qζ22qa)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)|32πδ, 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+δz2πδ, and

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

as z+ with 2π+δzπδ. 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|πδ.

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)nMn(j)(z,h),

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

28.20.17 Ien(z,h) =inMcn(1)(z,ih),
28.20.18 Ion(z,h) =inMsn(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) =eisπνMν(2)(z,h)+2icot(πν)sin(sπν)Mν(1)(z,h),
Mν(3)(z+sπi,h) =sin((s1)πν)sin(πν)Mν(3)(z,h)eiπνsin(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).