Even in a field such as elementary functions,
consistent treatment of mathematical properties makes it possible
to find new and meaningful results, many of which are presented
here for the first time, or to present new generalizations
of results that are already well-known.
Most of these functions have been verified using the command
FullSimplify on test equations. For example, evaluating
the formula Sin[2x]==2Cos[x]Sin[x] in Mathematica
returns the value True. When it is not possible to
verify the formulas algebraically, they have been verified
using carefully constructed methods. These methods take advantage
of Mathematica's knowledge of the properties of the
functions involved (branch cuts and singularities, for example)
to select test points, evaluate the expressions using Mathematica's
arbitrary-precision numerical capabilities, and then verify
A wide variety of methods were used to derive and verify
the identities listed on this site. Many standard techniques
were used to calculate indefinite and definite integrals.
Examples of other frequently used methods include symbolic
high-order series expansions and matching coefficients (for
instance, for deriving contiguous relations for generalized
hypergeometric functions and modular equations for modular
function), polynomialization and application of elimination
techniques (for instance, for deriving differential equations
for elliptic integrals and functions).
Along with its algebraic and numeric functionality, Mathematica
also includes a comprehensive suite of technical-publishing
and mathematical-typesetting capabilities. Every formula on
this site was typeset using Mathematica.
The HTML version of the website contains the identities
exported from Mathematica as GIFs, and the XML
the website contains the identities exported from
Since Mathematica notebooks are themselves Mathematica
expressions, all the content web pages on this website were
created using customized Mathematica programs. The
PDF files for each function were also automatically generated
All graphics shown in the Visualization section
website were calculated and rendered with Mathematica.
The complete source code for generating each graphic can be
downloaded from the corresponding function page.
Finally, because the Mathematica system is open-ended,
it has the potential for treating many more functions than
those listed here, and therefore it forms a good technical
basis for compiling knowledge not only about functions now
on this site but also about all mathematical functions.