・A Short History of Mathematical Function Tables
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 .
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. 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 entury.
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 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 tabulate.
・ベッセル関数
スイスの数学者・物理学者ダニエル・ベルヌーイ(Daniel Bernoulli, 1700年 - 1782年) によって定義され、ドイツの数学者・天文学者のフリードリッヒ・ベッセルによって一般化された関数。
物理で扱うベッセル関数は、電磁放射など、特定の振動数で同心円状または同心球状に広がる波動の方程式であるヘルムホルツ方程式の軸対称または点対称の系を取り扱うときに、半径方向の波の振動の解として現れることに由来する。
実際にベッセルが、惑星運動論において、太陽の周りを回る惑星の軌道を2体問題として扱うケプラー方程式を解くために、このベッセル関数を用いた。
(Wikipedia 参照)