Parabolic cylinder functions
Encyclopedia
In mathematics
, the parabolic cylinder functions are special functions defined as solutions to the differential equation
This equation is found, for example, when the technique of separation of variables
is used on differential equations which are expressed in parabolic cylindrical coordinates
.
The above equation may be brought into two distinct forms (A) and (B) by completing the square
and rescaling z, called H. F. Weber's equations :
(A)
and
(B)
If
is a solution, then so are
If
is a solution of equation (A), then
is a solution of (B), and, by symmetry,
are also solutions of (B).
(1965)):
and
where
is the confluent hypergeometric function
.
Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:
where
The function U(a, z) approaches zero for large values of |z| and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z .
and
For half-integer
values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials
; alternatively, they can also be expressed in terms of Bessel function
s.
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)):
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the parabolic cylinder functions are special functions defined as solutions to the differential equation
This equation is found, for example, when the technique of separation of variables
Separation of variables
In mathematics, separation of variables 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 on a different side of the equation....
is used on differential equations which are expressed in parabolic cylindrical coordinates
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...
.
The above equation may be brought into two distinct forms (A) and (B) by completing the square
Completing the square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...
and rescaling z, called H. F. Weber's equations :
(A)and
(B)If
is a solution, then so are
If
is a solution of equation (A), then
is a solution of (B), and, by symmetry,
are also solutions of (B).
Solutions
There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and StegunAbramowitz and Stegun
Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards...
(1965)):
and
where
is the confluent hypergeometric functionConfluent 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 singularity...
.
Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:
where
The function U(a, z) approaches zero for large values of |z| and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z .
and
For half-integer
Half-integer
In mathematics, a half-integer is a number of the formn + 1/2,where n is an integer. For example,are all half-integers. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer...
values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
; alternatively, they can also be expressed in terms of Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s.
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)):