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28 Mathieu Functions and Hill’s EquationNotation

§28.1 Special Notation

(For other notation see Notation for the Special Functions.)

m,n integers.
x,y real variables.
z=x+iy complex variable.
ν order of the Mathieu function or modified Mathieu function. (When ν is an integer it is often replaced by n.)
δ arbitrary small positive number.
a,q,h real or complex parameters of Mathieu’s equation with q=h2.
primes unless indicated otherwise, derivatives with respect to the argument

The main functions treated in this chapter are the Mathieu functions

ceν(z,q), seν(z,q), fen(z,q), gen(z,q), meν(z,q),

and the modified Mathieu functions

Ceν(z,q), Seν(z,q), Fen(z,q), Gen(z,q),
Meν(z,q), Mν(j)(z,h), Mcn(j)(z,h), Msn(j)(z,h),
Ien(z,h), Ion(z,h), Ken(z,h), Kon(z,h).

The functions Mcn(j)(z,h) and Msn(j)(z,h) are also known as the radial Mathieu functions.

The eigenvalues of Mathieu’s equation are denoted by

an(q),
bn(q),
λν(q).

The notation for the joining factors is

ge,n(h),
go,n(h),
fe,n(h),
fo,n(h).

Alternative notations for the parameters a and q are shown in Table 28.1.1.

Table 28.1.1: Notations for parameters in Mathieu’s equation.
Reference a q
Erdélyi et al. (1955) h θ
Meixner and Schäfke (1954) λ h2
Moon and Spencer (1971) λ q
Strutt (1932) λ h2
Whittaker and Watson (1927) a 8q

Alternative notations for the functions are as follows.

Arscott (1964b) and McLachlan (1947)

Feyn(z,q) =12πge,n(h)cen(0,q)Mcn(2)(z,h),
Men(1,2)(z,q) =12πge,n(h)cen(0,q)Mcn(3,4)(z,h),
Geyn(z,q) =12πgo,n(h)sen(0,q)Msn(2)(z,h),
Nen(1,2)(z,q) =12πgo,n(h)sen(0,q)Msn(3,4)(z,h).

Arscott (1964b) also uses -iμ for ν.

Campbell (1955)

inn =fen, cehn =Cen, inhn =Fen,
jnn =gen, sehn =Sen, jnhn =Gen.

Abramowitz and Stegun (1964, Chapter 20)

Fν(z)=Meν(z,q).

National Bureau of Standards (1967)

With s=4q,

Sen(s,z) =cen(z,q)cen(0,q),
Son(s,z) =sen(z,q)sen(0,q).

Stratton et al. (1941)

With c=2q,

Sen(c,z) =cen(z,q)cen(0,q),
Son(c,z) =sen(z,q)sen(0,q).

Zhang and Jin (1996)

The radial functions Mcn(j)(z,h) and Msn(j)(z,h) are denoted by Mcn(j)(z,q) and Msn(j)(z,q), respectively.