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Volume 9, Issue 1

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T R O T T ’ S C O R N E R
Michael Trott

What Information Is There?

Currently the website contains information about the approximately 250 mathematical functions of Mathematica. They are grouped into the following function categories. For a complete listing of the functions, visit the website.

Elementary • Constants • Bessel, Airy, Struve • Integer Functions • Polynomials • Gamma, Beta, Erf • Hypergeometric • Elliptic Integrals • Elliptic Functions • Zeta • Mathieu • Complex Components • Number Theory • Generalized Functions

As much as applicable, the available information is grouped into the following property categories.

Notations • Primary Definition • Specific Values • General Characteristics •  Series Representations • Integral Representations • Analytic Continuations • Product Representations • Limit Representations • Continued Fraction Representations • Generating Functions • Group Representations • Differential Equations • Difference Equations • Transformations • Operations (∂, ∫) • Integral Transforms (ℱ, ℋ, ℒ, ℳ) • Identities • Representations through More General Functions • Relations with Other Functions • Zeros • Inequalities • Theorems • Other Information • History and Applications • References

Each formula on this website is given a unique numeric identifier. As the site expands and more information is added, the identifier for a given formula will stay the same even if its position in a group changes. Consider this actual formula with identifier 01.06.27.0010.01:

(This formula demonstrates the previously mentioned branch cut topic. Most classic handbooks state but do not give a detailed description of for complex .)

Reading the identifier from the left, the groups of digits point to different types of information.

  • 01 stands for the function category of Elementary Functions.
  • 06 stands for the function Cos in the list of Elementary Functions.
  • 27 stands for the property category Representations through Equivalent Functions.
  • 0010 numbers the formula within this section.
  • 01 means the modification number of this formula.

Let me give some more examples of formulas found on the website.

The next formula describes the direction-dependent asymptotics of the Airy function .

The indefinite integral of a Bessel function.

A closed-form formula for the th prime number.

In addition to formulas and identities for mathematical functions, theorems about them and general mathematical identities are shown and are downloadable from the website.

What New Information Is There?

Thousands of the formulas and identities on the website are new and were derived especially for the website. With Mathematica’s abilities to return special functions (Integrate, Sum, DSolve), to simplify them (FullSimplify, FunctionExpand), and to numericize them (N), Mathematica is an ideal environment to verify identities and to derive new ones. Let me give one example here: the addition formula for the Weierstrass elliptic function . In handbooks one finds the formula

In this form the formula is unsatisfying because the right-hand side contains the function in addition to . To derive a formula only in , I supplement the given formula with the three differential equations for , , and and generate four polynomial equations by clearing the denominators.

Using GroebnerBasis with EliminationOrder allows us to eliminate the s after a few minutes on a fast machine.

The result is a polynomial of total degree 18 with 670 terms.

Here are the first 25 terms of the polynomial.

The double-argument specification yields another previously unknown formula.

Here is another small example: a short program that derives the contiguous relation for Gauss’ hypergeometric function. It takes about an hour on a fast machine.

While these identities are all well known, generalizing this approach allowed us to derive the contiguous relations for various functions that were previously not known.



     
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