Friday, March 1, 2013

Iteration types of differentiation

The differentiation operation is a transformation of differentiable functions and it has different iteration types for different differentiable functions:
  • Identity: $c_1e^x$
  • Idempotent: $c_1e^x + c_2$
  • Involution: $c_1e^x + c_2e^{-x}$
All functions of a certain iteration type come from solutions to first order linear ordinary differential equations such as $y'=y$, $y''=y$, and $y''=y'$. The partial ordering of iteration types helps us to understand these differential equations. In general, polynomials, exp, sin, cos, sinh, and cosh and other similar functions all produce a finite number of differentiations.

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