# Complexity of analytic functions and integrals

There exist polynomial - time computable functions, log - space computable functions, and NC - functions. Given this:

To which class do analytic elementary functions, including trigonometric ones, belong? To which class do analytic special functions belong? Is the evaluation of their integrals polynomial-time computable, i.e., do they belong to the class P

• integration of polynomial time computable real functions is PSpace complete. Commented Oct 12, 2023 at 12:04
• Ker I. Ko's book is a good read on the topic of complexity of real functions. Commented Oct 12, 2023 at 12:05
• www3.cs.stonybrook.edu/~keriko/cca10.pdf Commented Oct 13, 2023 at 0:46

You can compute approximations of elementary analytic functions to any given precision on rational inputs in the function class corresponding to uniform $$\mathrm{TC}^0$$ (and a fortiori in FP), as long as you take care that the output is not too large: i.e., in the case of $$\exp$$, the inputs are from a bounded domain so that the output cannot get exponentially large.
The same should hold for typical special functions, but this is an open-ended designation so it can't be answered definitively without specifying exactly what functions you mean. Generally speaking, analytic functions given by power series whose coefficients are $$\mathrm{TC}^0$$-computable (which any simple enough "formula" involving polynomials, powers, fractions, factorials, and the like is) can be approximated in $$\mathrm{TC}^0$$ on any compact subset of their domain of convergence.