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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.5 Second Solutions fen, gen


§28.5(i) Definitions

Theorem of Ince (1922)

If a nontrivial solution of Mathieu’s equation with q0 has period π or 2π, then any linearly independent solution cannot have either period.

Second solutions of (28.2.1) are given by

28.5.1 fen(z,q)=Cn(q)(zcen(z,q)+fn(z,q)),

when a=an(q), n=0,1,2,, and by

28.5.2 gen(z,q)=Sn(q)(zsen(z,q)+gn(z,q)),

when a=bn(q), n=1,2,3,. For m=0,1,2,, we have

28.5.3 f2m(z,q)π-periodic, odd,f2m+1(z,q)π-antiperiodic, odd,


28.5.4 g2m+1(z,q)π-antiperiodic, even,g2m+2(z,q)π-periodic, even;

compare §28.2(vi). The functions fn(z,q), gn(z,q) are unique.

The factors Cn(q) and Sn(q) in (28.5.1) and (28.5.2) are normalized so that

28.5.5 (Cn(q))202π(fn(x,q))2dx=(Sn(q))202π(gn(x,q))2dx=π.

As q0 with n0, Cn(q)0, Sn(q)0, Cn(q)fn(z,q)sinnz, and Sn(q)gn(z,q)cosnz. This determines the signs of Cn(q) and Sn(q). (Other normalizations for Cn(q) and Sn(q) can be found in the literature, but most formulas—including connection formulas—are unaffected since fen(z,q)/Cn(q) and gen(z,q)/Sn(q) are invariant.)

28.5.6 C2m(-q) =C2m(q),
C2m+1(-q) =S2m+1(q),
S2m+2(-q) =S2m+2(q).

For q=0,

28.5.7 fe0(z,0) =z,
fen(z,0) =sinnz,
gen(z,0) =cosnz,

compare (28.2.29).

As a consequence of the factor z on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as z± on .


28.5.8 𝒲{cen,fen} =cen(0,q)fen(0,q),
28.5.9 𝒲{sen,gen} =-sen(0,q)gen(0,q).

See (28.22.12) for fen(0,q) and gen(0,q).

For further information on Cn(q), Sn(q), and expansions of fn(z,q), gn(z,q) in Fourier series or in series of cen, sen functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72).

§28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation

Odd Second Solutions

See accompanying text
Figure 28.5.1: fe0(x,0.5) for 0x2π and (for comparison) ce0(x,0.5). Magnify
See accompanying text
Figure 28.5.2: fe0(x,1) for 0x2π and (for comparison) ce0(x,1). Magnify
See accompanying text
Figure 28.5.3: fe1(x,0.5) for 0x2π and (for comparison) ce1(x,0.5). Magnify
See accompanying text
Figure 28.5.4: fe1(x,1) for 0x2π and (for comparison) ce1(x,1). Magnify

Even Second Solutions

See accompanying text
Figure 28.5.5: ge1(x,0.5) for 0x2π and (for comparison) se1(x,0.5). Magnify
See accompanying text
Figure 28.5.6: ge1(x,1) for 0x2π and (for comparison) se1(x,1). Magnify