ABSTRACT
The analytic continuation (Chapter 31) allows the study of a complex function over the whole of the complex plane, including the neighborhood of singularities (Section 39.1); the latter include poles (essential singularities) where the function is infinite (indeterminate), that is, exceeds any bound (Section 39.1) [(comes arbitrarily close to any value, however, small or large (Section 39.2)]. This result can be extended to the Picard second theorem (Section 39.3) stating that an integral function takes all values an infinite number of times, possibly excluding one exceptional value; the exponential gives the example of an exceptional value, viz. it is never zero, so exp(z) = 0 has no roots and exp(z) = c has an infinite number of roots for all c = 0 ≡ b. This demonstrates the existence of an infinite number of roots for any transcendental complex equation f (z) = c, where f (z) is an analytic nonrational function, for all values of c, excluding possibly one exceptional value b = 0, viz. b = 0 for the exponential f (z) ≡ expz. The first Picard theorem states that an analytic function that has two exceptional values is a constant (Section 39.3); this is a considerable extension of the Liouville theorem (Section 27.6) since it shows that an analytic function reduces to a constant if it fails to take any two distinct values, that is, f(z) = a, b with a = b implies f(z) = const. The two theorems of Picard can be proved via the elliptic modular function of Legendre that relates to coverings of the complex plane by Schwartz triangles (Section 39.4). The first Picard theorem states that a function that does not take two values is a constant, and holds not only over the whole complex plane but also over parts of it, viz.: (i) over a dense ray, that is, an infinitesimal angular sector where, by the Julia theorem (Section 39.5), lie an infinity number of roots of a transcendental equation for every nonexceptional value; (ii) over a disk, with the Landau radius (Section 39.6) that is the maximum value of the radius beyond that the function either ceases to holomorphic or takes all values but one an infinite number of times. These theorems apply to nonrational analytic functions, that is, to transcendental functions. An important class of transcendental functions is related to hyperelliptic integrals that are integrals of the inverse square root of a polynomial, with branch-cuts joining its roots. The integrals along these branch-cuts specify the periods (Section 39.7), that is, quantities that may be added to the integral without changing the inverse function. Considering polynomials of the second degree leads to (Section 39.8) the periods of the exponential, circular, and hyperbolic functions. The polynomials of third (fourth) degree lead (Section 39.9) to the elliptic functions of Weierstrass (Jacobi). The assurance given by the rather abstract theorems of Casorati-Weierstrass and Picard that every complex nonalgebraic or transcendental equation has roots, actually underlies the feasibility of the numerical methods of computation of the roots.
