In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the parabolic cylinder functions are
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 by ...
s defined as solutions to the differential equation
This equation is found when the technique of
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
is used on
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
when expressed in
parabolic cylindrical coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the
perpendicular z-direction. Hence, the coordinate surfaces ...
.
The above equation may be brought into two distinct forms (A) and (B) by
completing the square
:
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
:ax^2 + bx + c
to the form
:a(x-h)^2 + k
for some values of ''h'' and ''k''.
In other words, completing the square places a perfe ...
and rescaling , called
H. F. Weber
Heinrich Friedrich Weber (; ; 7 November 1843 – 24 May 1912) was a physicist born in the town of Magdala, Germany, Magdala, near Weimar.
Biography
Around 1861 he entered the University of Jena, where Ernst Abbe became the first of two phys ...
's equations:
and
If
is a solution, then so are
If
is a solution of equation (), then
is a solution of (), and, by symmetry,
are also solutions of ().
Solutions
There are independent even and odd solutions of the form (). These are given by (following the notation of
Abramowitz and Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
(1965)):
and
where
is the
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
.
Other pairs of independent solutions may be formed from linear combinations of the above solutions.
One such pair is based upon their behavior at infinity:
where
The function approaches zero for large values of and , while diverges for large values of positive real .
and
For
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
values of ''a'', these (that is, ''U'' and ''V'') can be re-expressed in terms of
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well a ...
; alternatively, they can also be expressed in terms of
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s.
The functions ''U'' and ''V'' can also be related to the functions (a notation dating back to Whittaker (1902))
that are themselves sometimes called parabolic cylinder functions:
Function was introduced by Whittaker and Watson as a solution of eq.~() with
bounded at
.
It can be expressed in terms of confluent hypergeometric functions as
:
Power series for this function have been obtained by Abadir (1993).
References
{{DEFAULTSORT:Parabolic Cylinder Function
Special hypergeometric functions
Special functions