What is special about special functions?
Just taking a look at their
representations makes one understand that these functions make up a
select set in the world of functions: their series coefficients and continued
fraction elements are all known through simple explicit formulas which in
addition exhibit some nice monotonicity properties.
What is special about continued fractions?
On the one hand continued
fraction representations of functions enjoy larger convergence domains
than their series counterparts, while on the other hand they are equally simple
to deal with. It suffices to build some understanding for them by reading
part I of the handbook.
What is unique about this project?
Books on special functions do
mostly not contain continued fraction representations. Books on continued
fractions occasionally serve up some special function as example. This
handbook is the result of a systematic study of continued fraction
representations of special functions. Only 10% of the continued fractions
in part III can also be found in the Abramowitz and Stegun handbook or at
special functions websites. The project is still ongoing and more chapters
and functions will be added in the future.
How trustworthy is the compendium of formulas?
Those formulas from the
handbook which are implemented in our Maple library have been validated in
two ways. They were automatically transferred from text to program
to allow the detection of printing errors. They were then compared to
existing implementations for the functions they represent. Any errors
found in original reference material is thereby corrected.
Which software is developed in the wake of this encyclopedic study?
Essentially three tools are made available with an accompanying web
interface. All tables printed in the handbook can be tailor made
interactively: one can put more formulas in the side by side comparison,
choose higher or lower order approximants of the representations, or
select different sample values for the arguments and parameters.
All series and continued fraction representations of the elementary and
special functions are preprogrammed in a downloadable Maple library,
developed especially to offer the functionality required for handling
(limit k-periodic) continued fractions. Easy truncation and roundoff error
bounds for the series and continued fraction representations have allowed to
develop a validated scalable precision (up to a few thousand digits) and
multiradix (powers of 2 or 10) numeric C++ library for the evaluation of
these special functions (available in the near future).
J. Van Deun
W. B. Jones
- Table of Contents
- Chapter 0
- download CFSF Maple library
- tools used in the implementation
- appendix with code extract
DLMF project: a revision of the
Handbook of mathematical functions with formulas, graphs and mathematical tables (Abramowitz and Stegun)
Askey-Bateman project: a revision of the
Bateman Manuscript Project (A. Erdélyi et al.)
More information: [SIAM News], [OP-SF NET]