## Thursday, June 14, 2012

### Arithmetic module

The arithmetic module deals with functions that are built out of the of addition, multiplication, and composition. All of these higher order operations have derivative laws associated with them: the sum rule, the product rule, and the chain rule. As such, this module includes derivative operations.

Entire functions are described entirely using power series. The functions exp, sin, cos, sinh, cosh, erfi, the reciprocal of gamma, all polynomials, and all addition, multiplication, and compositions of entire functions are entire. However, entire functions are not closed under inversion or division so ln, sqrt, and tan are not entire.

Linear ordinary differential operators receive functions as arguments and then they output new functions using higher order operations and differentiation. Linear ordinary differential operators are closed under composition, so they form their compositional monoid. The derivative function itself is a differential operator that uses a hidden multimethod internally that dispatches on the type of its arguments.