10217

# 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.

### DETAILS

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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.