This is mainly a question about the remainder term of power series for elementary functions.

I'm very interested in aspects of calculating or computing elementary operations and functions, by which I mean:

- trigonometric: $\sin$, $\cos$, $\tan$
- inverse trig.: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$
- log and exponential: $\ln$, $\exp$
- hyperbolic: $\sinh$, $\cosh$, $\tanh$
- inverse hyp.: $\sinh^{-1}$, $\cosh^{-1}$, $\tanh^{-1}$
- powers, reciprocation, $\sqrt{\ \ \ }$

perhaps also:

- gamma function: $\Gamma$
- and a few other important functions

There are many contexts (of calculation). For example:

- real
*versus*complex arguments - known, fixed precision
*versus*variable precision - numerical
*versus*symbolic

There are many approaches and techniques available too. For example:

- power series expansions and polynomial approximations
- use of relationships between the functions
- use of periodic or similar properties to shrink the domain
- lookup tables and interpolation
- CORDIC (used within some hand calculators I believe)
*exact*methods- interval or other error-tracking methods

Some good references to certain aspects include:

- Digital Library of Mathematical Functions: Elementary Functions
- Chee-Keng Yap, Fundamental problems of algorithmic algebra
- Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Designs

**The main gap in my knowledge is in finding bounds for the error or remainder term in partial power series expansions of certain of the above functions. Some are fairly simple to determine, whilst others seem to be awkward.**

**Any pointers on this matter would be much appreciated.**

Likewise for any further references on any other aspects of or techniques for calculating elementary functions.

detailedquestion would be helpful; can you give a particular power series for which you cannot determine a good estimate for the remainder, by any of the standard methods you've listed? $\endgroup$amateurmathematician. As a particular case: the terms of the expansion of sin are simple because the sequence of derivatives remains simple. On the other hand, the terms of the expansion of tan seem to become very unwieldy; the closed form is superficially simple, but involves the Bernoulli numbers, which are somewhat erratic. I expect I'm missing something quite simple. In saying that, I wasn't fishing for a specific solution so much as someleads. Thanks again. $\endgroup$2more comments