In the physical sciences, the Airy function (or Airy function of the first kind) is a
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
named after the British astronomer
George Biddell Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
(1801–1892). The function and the related function , are linearly independent solutions to the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
known as the Airy equation or the Stokes equation. This is the simplest second-order
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
with a turning point (a point where the character of the solutions changes from oscillatory to exponential).
Definitions
For real values of ''x'', the Airy function of the first kind can be defined by the
improper Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
:
which converges by
Dirichlet's test
In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the ''Journal de Mathématiques Pures et Appliquées'' in 186 ...
. For any real number
there is positive real number
such that function
is increasing, unbounded and convex with continuous and unbounded derivative on interval