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algebraic Amer asymptotic behavior automorphic functions Bessel functions called Chapter characteristic functions characteristic values circle coefficients computation const constant continued fraction convergence cosh curvilinear coordinates defined denoted determined differential equation ellipsoidal wave functions Erde'lyi expansions finite fixed point follows formulas Fricke func fundamental region half-plane Heun's equation homographic hypergeometric imaginary period Ince infinite integer integral equations investigated Lame Lame's Lamp's equation Laplace's equation Legendre functions London Math maps Mathieu functions Mathieu's equation McLachlan 1947 Meixner modified Mathieu functions modular group multiple notation obtained odd function orthogonal period 2K periodic solutions polynomials of Chap positive integer power series Proc quadratic Ramanujan real period recurrence relations satisfy second kind sections singular sinh solution of Mathieu's spheroidal wave functions Strutt substitutions surface harmonics theorems theory third kind tions transcendental transformation trigonometric values of h Whittaker z-plane zeros