Question

Where can I turn for information regarding computing times of mathematical functions? Has any (general) study with any amount of rigor been made?

For instance, the computing time of

constant + constant

generally takes O(1).

Suppose I want to start using math like integrals, and I’d like to get an asymptotic approximation to various integrals. Has there been a standard study of this, or must I take the information I have and figure out my own approximation. I’d be very interested in a standard approach to this, and I’d like to know if it already exists.

Here’s my motivation:
I’m in the middle of writing a paper that points out the equivalence between NP hard problems and certain types of mathematical equations. It seems that there might be use for a study of math computing times that is generalized like a new science.

EDIT:
I guess I’m wondering if there is a standard computational complexity to any given math that cannot be avoided. I’m wondering if anyone has studied this question. I’d love to see what others have tried.

EDIT 2:
Wikipedia lists “Computational Complexity Theory” in their encyclopedia, which I think may fit the bill. I’m still wondering if someone who has studied this could affirm this.

Answer 1

There isn’t a collected body of work, but work on approximating functions comes close. For example, you’d like to know that approximating sin(x) to within an epsilon error can be done in time proportional to some polynomial in log(x) and 1/epsilon. There isn’t a general theory here (you should look up information complexity though), and focusing on specific functions might help.