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}.
(d^2)u / (dr)^2 + du / (r * dr) + a^2 - b^2 / r^2 = 0 {Bessel equation}. x^2 * y'' + x * y' + (x^2 - n^2) * y = 0, where (d^2) is second derivative and x and n are complex, has two solutions. J(n,x) = (1 / (2 * pi)) * (integral from u = 0 to u = 2*pi of (cos(n*u) - x * sin(u)) * du). x * J(n+1,x) - 2 * n * J(n,x) + x * J(n-1,x) = 0.
3-Calculus-Differential Equation-Kinds
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Date Modified: 2022.0225