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Even in a field such as elementary functions, Mathematica's 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 agreement numerically.

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 version of the website contains the identities exported from Mathematica as MathML.

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 using Mathematica.

All graphics shown in the Visualization section of this 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.



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