Boole Differential Equation with Continued Fractions
Explore the solutions of the Boole differential equation with continued fractions. Continued fractions provide a very effective function approximation toolset. Usually the continued fraction expansion of a given function approximates the function better than the Taylor series or the Fourier series. The solution(s) of the Boole differential equation are very diverse; they contain polynomials, trigonometric functions, hyperbolas, (nested) square roots, and modular forms.
The solutions of the Boole differential equation are ratios of Bessel functions. In many cases these simplify to monomials, modular forms, variations of trigonometric functions, and logarithms. The number of terms used in the continued fraction expansion is , is the domain on the axis ("x range"), and is the range of values displayed on the axis ("y range").
Snapshot 1: gamma = 0, --> monomials ( is the exponent). No residual error.
Snapshot 2: alpha = 0, --> modular forms ( is the exponent of the term under the square root). There are always two solutions.
Snapshot 3: Bessel functions simplify to exponential functions.