In the 18th and 19th centuries, scientists discovered that the elementary functions—powers, roots, trigonometric
functions and their inverses—had their limitations. They found that solutions for some important physical
problems—like the orbital motion of planets, the oscillatory motion of suspended chains and the calculation of the
gravitational potential of nearly spherical bodies—could not always be described in a closed form using only elementary
functions. Even in the realm of pure mathematics, some quantities—such as the circumference of an ellipse—were also
impossible to discuss in such terms. Functions describing solutions to these problems were often expressed as infinite
series, as integrals or as solutions to differential equations.
On further investigation, scientists noted that a relatively small number of these special functions turned up
over and over again in different contexts. What's more, they noted that many other problems could be solved in the form
of a comparatively simple combination of these newer functions with the elementary functions known to the ancients.
Functions that cropped up most frequently in scientific calculations were given names and notations that have come into
common usage: Bessel functions, Struve functions, Mathieu functions, the spherical harmonics, the Gamma function, the
Beta function, Jacobi functions and most of the others appearing on this website.
In the second half of the 19th century mathematicians also started to investigate these special functions from a purely
theoretical perspective. Alternative representations—as differential equations, series, integrals, continued fractions
or other forms—were found for many. Important publications on the topic at the turn of the century include the
four-volume masterpiece on the elliptic functions by J. Tannery and J. Molk (published 1893–1902), containing hundreds
of pages of collected formulas; I. Todhunter's treatise on Laplace, Lamé and Bessel functions (1875); E. Heine's
treatise on spherical harmonics (1881); and A. Wangerin's work on the same topic (1904).
Large tables with numerical values for the special functions also began to appear, along with three-dimensional "graphs"
made of wood or plaster—masterpieces of precision sculpting—showing the behavior of functions such as P and the Jacobi
functions. Many of these models are still on display in math departments throughout the world and the graphics on this
website can be thought of as their modern, computer-drawn counterparts.
Charles Babbage, who designed but was unable to build the "difference engine," planned a printing device allowing the
machine to generate large tables automatically. A Swedish publisher named Georg Scheutz and his son Edvard later built a
difference engine that could set type. In 1857, the Scheutzes produced a mechanically generated table of common
logarithms to five decimal places for the integers from 1 to 10,000; each value took about thirty seconds to calculate.
Funktionentafeln mit Formeln und Kurven, the first modern handbook of special functions—that is, one containing
graphs, formulas and numerical tables—was published in 1909 by Eugene Jahnke and Fritz Emde. The first text dealing
comprehensively with most of the named special functions was E. T. Whittaker and G. N. Watson's A Course of Modern
Analysis, 2nd Edition (1915). This popular text consisted of two parts; Part I is a textbook of complex analysis,
while Part II is a handbook of special functions.
In 1939, orthogonal polynomials were given their first detailed treatment by Gábor Szegö. This work was followed in 1943
by Wilhelm Magnus and Fritz Oberhettinger's Formeln und Lehrsätze für die speziellen Funktionen der Mathematischen
Physik, the most complete collection of formulas involving special functions yet prepared.
The massive Bateman Manuscript Project—the editing for posthumous
publication of Harry Bateman's accumulated notes on
special functions, which he stored in shoe boxes—was carried out by Arthur Erdélyi, Wilhelm Magnus, Fritz
Oberhettinger and Francesco Tricomi, culminating in 1953 with the classic three-volume work Higher Transcendental
Functions. This monumental collection contains not only formulas, but also derivations, proofs and historical
remarks. (Along with the contents of Higher Transcendental Functions, Bateman's shoe boxes held enough material
for a two-volume set of integral transform tables.)
In Higher Transcendental Functions, Erdélyi introduced a new emphasis on the importance of hypergeometric
functions as an underlying, unifying basis for the development of many of the categories of special functions. Although
2F1 had been extensively studied since the time of Gauss, mathematicians were slow to appreciate
the importance of the generalized hypergeometric pFq and to recognize the many relations between
hypergeometric functions and the special functions encountered most frequently. Another innovation in Higher
Transcendental Functions was the inclusion of number-theoretical functions.
Parallel to the handbooks dealing with series expansions, differential equations, functional identities and so forth of
special functions, many integral tables were developed in the 20th century. Among these, especially the tables of W.
Gröbner, N. Hofreiter, A. Erdelyi, W. Magnus, F. Oberhettinger, I. S. Gradshteyn, I.H. Ryshik, H. Exton, H. M.
Srivastava, A. P. Prudnikov, Ya. A. Brychkov and O. I. Marichev are noteworthy.
Application of the electronic computer resulted in many massive volumes containing hundreds of pages of tables for
Bessel functions, elliptic integrals, Legendre functions and so on. An important handbook containing graphs, formulas,
and compute-generated numerical data was assembled by Milton Abramowitz and Irene Stegun. This work was published in
1964 by the National Bureau of Standards as the Handbook of Mathematical Functions with Formulas, Graphs and
Mathematical Tables. Individual chapters were compiled by various authors, leading to a certain unevenness in the
quality of the material and its presentation. Nevertheless, the Handbook of Mathematical Functions remains a
standard reference and is still in widespread use.
Ironically, the computer that led to the creation of such mammoth numeric tables is now eliminating the need for them.
The ready availability of computer processing time and technical software now allows technical users to calculate the
values of any needed function without recourse to reference works. Mathematica can calculate every special
function on this website to any desired precision for any real or complex values of the arguments and parameters.
Additionally, Mathematica can symbolically and numerically calculate values for integrals or other operations and
transformations involving these functions, providing far more information than any single handbook could possibly
Thereby the need for an even more comprehensive collection of special functions persists.
And the Mathematical Functions Site is the most complete such resource today, with tens of thousands of identities,
some extending over multiple pages if printed. The website is virtually arbitrarily extensible and not bound to the limitations of a printed
book. Updating is easy, so new information can be quickly incorporated into the website.