ABSTRACT
Concerning the six circular (hyperbolic) functions, namely, the two primary functions sine and cosine, and the four secondary functions secant, cosecant, tangent, and cotangent, the algebraic (analytic) properties are discussed in Chapter 5 (Chapter 7). The direct functions are periodic and hence have as inverses the 12 cyclometric functions that are many-valued and can be made single-valued by choosing the principal branch in a complex plane with suitable branch cuts (Section 7.2). Concerning infinite representations: (1) the four primary functions are integral, with an infinite number of zeros, and thus can be represented by power series with infinite radius of convergence (Section 7.1), by infinite products (Section 7.7), and by continued fractions (Section 7.8); (2) the eight secondary functions are meromorphic and can be represented by power series with finite radius of convergence (Section 7.1), by series of fractions (Section 7.6), and by continued fractions (Section 7.8); and (3) the cyclometric functions have branch points (branch cuts) that limit the radius of convergence (domain of validity) of their representation by power series (Section 7.4) [continued fractions (Section 7.8)]. In the process of establishing these infinite representations: (1) the Euler and Bernoulli numbers are introduced to specify the coefficients of the power series for secondary functions (Section 7.1); (2 and 3) the zeros and slopes (poles and residues) lead (Section 7.5) to infinite products (series of fractions) for the primary (secondary) functions [Section 7.7 (Section 7.6)]; (4) the derivatives and primitives (Section 7.3) specify the coefficients in the power series for the direct (inverse), that is, primary and secondary (cyclometric) functions [Section 7.1 (Section 7.4)]; (5) the continued fractions (Section 7.8) are obtained by transformation [Subsections 1.8.1 and 1.8.2 (Subsections 1.8.4 and 1.8.5)] of power series (infinite products) or by the Lambert method (Section 1.9) applied to functions of the hypergeometric family (Section 5.9); and (6) the power series, infinite products, and continued fractions can be used to calculate irrational numbers like e (π), that are the base of natural logarithms (Subsection 1.8.5 and Section 3.9) [solve the quadrature of the circle (Subsection 1.8.7 and Section 7.9)].
