Circular, elliptic, hyperbolic, and other analytic functions {automorphic function} can generalize to find higher properties.
invariance
Automorphic functions are invariant if z' = (a*z + b) / (c*z + d) where a*d - b*c = 1, z is complex number, and z' is complex conjugate.
theta
theta(z) = sum from i = 0 to i = infinity of (c(i) * z + d(i))^(-2 * m) * H(z(i)), where m > 1 and H is rational function. Automorphic-function groups can be discrete or discontinuous groups of infinite order {theory of automorphic functions, discontinuous} {automorphic function theory, discontinuous}.
Mathematical Sciences>Calculus>Differential Equation>Kinds>Complex
3-Calculus-Differential Equation-Kinds-Complex
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Date Modified: 2022.0224