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

§28.7 Analytic Continuation of Eigenvalues

As functions of q, an(q) and bn(q) can be continued analytically in the complex q-plane. The only singularities are algebraic branch points, with an(q) and bn(q) finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. In consequence, the functions can be defined uniquely by introducing suitable cuts in the q-plane. See Meixner and Schäfke (1954, §2.22). The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of q are normal values. To 4D the first branch points between a0(q) and a2(q) are at q0=±i1.4688 with a0(q0)=a2(q0)=2.0886, and between b2(q) and b4(q) they are at q1=±i6.9289 with b2(q1)=b4(q1)=11.1904. For real q with |q|<|q0|, a0(iq) and a2(iq) are real-valued, whereas for real q with |q|>|q0|, a0(iq) and a2(iq) are complex conjugates. See also Mulholland and Goldstein (1929), Bouwkamp (1948), Meixner et al. (1980), Hunter and Guerrieri (1981), Hunter (1981), and Shivakumar and Xue (1999).

For a visualization of the first branch point of a0(iq^) and a2(iq^) see Figure 28.7.1.

See accompanying text
Figure 28.7.1: Branch point of the eigenvalues a0(iq^) and a2(iq^): 0q^2.5. Magnify

All the a2n(q), n=0,1,2,, can be regarded as belonging to a complete analytic function (in the large). Therefore wI(12π;a,q) is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). Analogous statements hold for a2n+1(q), b2n+1(q), and b2n+2(q), also for n=0,1,2,. Closely connected with the preceding statements, we have

28.7.1 n=0(a2n(q)(2n)2) =0,
28.7.2 n=0(a2n+1(q)(2n+1)2) =q,
28.7.3 n=0(b2n+1(q)(2n+1)2) =q,
28.7.4 n=0(b2n+2(q)(2n+2)2) =0.