
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.