The Mathematica Journal: Volume 9, Issue 1: T R O T T ’ S C O R N E R

T R O T T ’ S C O R N E R

Michael Trott

Why Care about Mathematical Functions?

Special functions occur in exact models and approximate solutions of virtually thousands of problems in mathematics and the exact sciences. Typical examples are the eigenfunctions of a quantum mechanical harmonic oscillator that contain Hermite polynomials, or Bessel functions that appear in the solution of Newton’s equations of motion for a chain of masses connected by springs. There are so many examples that I do not even attempt to give references at this point. The fact that these functions were given a name indicates their role in the exact sciences better than anything else.

Mathematica users are surely used to a variety of special functions returned by functions such as Integrate, Sum, and DSolve. Here are some typical examples.

Why Make Such a Website at All?

While Mathematica is able to perform a variety of operations on special functions, it does not completely replace special functions textbooks and reference books. Here is a list of the main reasons why such a website is still needed at the beginning of the twenty-first century.

The classic handbooks about special functions (Abramowitz/Stegun, Bateman/Erdelyi) are slightly out of date. Many new results have been found for many special functions since their publication in 1964 and 1950–54 respectively.

Not all functions used in today’s research are covered in these classic handbooks (e.g., the polylogarithm and Meijer G-functions).

There is no longer any need for tables of numerical values due to the built-in, high-precision numeric capabilities of modern software for all complex values of all special functions.

The role of special functions in modern science is continually increasing. For instance, the number of mathematics, physics, and engineering articles dealing with special functions increases by about eight percent per year.

Although the choice of branch cuts is largely arbitrary, in practice, people have adopted a small set of conventions. In computer mathematics all branch cuts of all functions follow uniquely from the branch cut of log or power. To maintain consistency, some classical formulas for special functions must be rewritten.

Collecting this material is necessary because we still lack a universal set of algorithms to generate it. Encouraging progress has been made, for instance, in algebraic summation and the integration of elementary functions.

Many important series can only be described by programs, for which no closed form for the nth term exists (e.g., zeros of Airy functions).

The material is not available in a computer-readable form.

The classical handbooks are not uniform in style and organization. (Their various chapters were compiled by different authors with different criteria for inclusion of material.) Clearly, the ideal form would be a giant table.