(2.1)(2.1)

Lecture 19: Ordinary Differential Equations: Special Functions

Key pointsHermite differential equation: Legendre's differential equation: Bessel's differential equation: Modified Bessel differential equation: Spherical Bessel differential equation: Airy differential equation: Laguerre differential equation:

MapleHermiteH(n,x)LegendreP(n,x), LegendreQ(n,x)BesselJ(n,x), BesselY(n,x)HankelH1(n,x), HankelH2(n,x)BesselI(n,x), BesselK(n,x)AiryAi(x), AiryBi(x)LaguerreL(n,x)KummerM(n,x), KummerU(n,x)

Hermite equation

Hermite differential equation

General solution by Maple

where and are the Kummer functions . The Kummer functions are also calledconfluent hypergeometric functions.In Maple, they are predifined functions, and .

The two independent solutions for the Hermite differential equation is and

(2.3)(2.3)

.

For integer , Hermite polynomial

is a solution to the Hermite differential equation. .

For ,

In Maple, Hermite polynomials are predefined as HermiteH(n,x)The first few Hermite polynomials are:

= = simplify

1

= = simplify

= = simplify

= = simplify

The Hermite polynomials determined by the following recursive relation are solution to the

Hermite equation.

Orthogonality

The second solution to the Hermite equation is the second kind Hermite function which

exponentially diverges as . Since it is rarely used in physics, we don't discuss it here.

Legendre equation

Legendre's differential equation of degree n (0th order)

(3.1)(3.1)

(3.2)(3.2)

General Solution

For ,

Two linearly independent solutions to this ODE is known as the first kind of Legendre polynomials and the second kind of Legendre function . is not popular in Physics because it is

defined for and . (It is possible to extend to but it is not our

interest.)

First kind Legendre polynomials

In Maple, Legendre polynomials are predefined as .

= 1 = x

= = simplify

= = simplify

= = simplify

= = simplify

= = simplify

Note that diverges logarithmically at and .

Orthogonality

forms an orthonormal basis set for .

Recursive equation

General Legendre equation

Legendre's differential equation

where and are integers and . (mathematically speaking non-integer values are allowed but not popular in physics.)Associate Legendre functions,

are solution to the general Legendre differential equation.

= = simplify

= = simplify

= = simplify

= = simplify

= = simplify

= = simplify

1

Solution by Maple

(5.2)(5.2)

(4.2)(4.2)

(4.1)(4.1)

(5.1)(5.1)

Bessel equation

Bessel's differential equation

General solution by Maple

Two linearly independent solutions are known as Bessel function, and Weber function .

The second solution is also called Neumann function and denoted as .

Hankel functions are also a pair of linearly

independent solutions.Even for integer , there is no simplex expression: For ,

In Maple, these functions are predefined as BesselJ(n,x), BesselY(n,x), BesselH1(n,x), and BesselH2(n,x). These functions can be expressed only with infinite series (Maple cannot express them in simple forms but you can evaluate numerical values with Maple.)

(4.1)(4.1)

(6.1)(6.1)

Bessel functions are not orthogonal!

Bessel function Weber function

Modified Bessel equation

Modified Bessel differential equation

General solution by Maple

Two linearly independent solutions are the first kind and second kind of modified Bessel functions, and , respectively.

In Maple, the modified Bessel functions are predefined as BesselI(n,x) and BesselK(n,x). There is no

(6.2)(6.2)

(4.1)(4.1)

simple expression of the modified Bessel functions even for integer .

The modified Bessel functions are related to the regular Bessel functions as follows:

In Maple, the modified Bessel functions are predefined as BesselI(n,x) and BesselK(n,x). There is no simplex expression for these functions.

Spherical Bessel equation

Spherical Bessel differential equation

General Solution

(7.2)(7.2)

(4.1)(4.1)

(7.1)(7.1)

Two linearly independent solutions are spherical Bessel functions:

Although there is no simple expression of Bessel functions in general, the spherical Bessel functions can be written in simple closed form:For ,

For general integer ,

Spherical Bessel functions can be expressed in simple form. For example,

= = simplify symbolic

x

= = simplify symbolic

x2

(4.1)(4.1)

(7.1)(7.1)

= = simplify symbolic

= = simplify symbolic

Note that the spherical Neumann functions diverge at .

Spherical Bessel Spherical Neumann function

Airy equation

Airy differential equation

General Solution

(8.1)(8.1)

(9.1)(9.1)

(7.1)(7.1)

(4.1)(4.1)

Two linearly independent solutions are the first and second kind of Airy functions, Ai(x) and Bi(x), respectively. They are related to modified Bessel functions as follows:

Since , the second term is usually eliminated by physical boundary condition.In Maple, the Airy functions are predefined as AiryAi(x) and AiryBi(x).

1st kind of Airy function, Ai(x) 2nd kind of Airy function, Bi(x)

Laguerre equation

Laguerre differential equation

General Solution

The Kummer functions are the two independent solutions for the Laguerre equation.

(9.2)(9.2)

(8.1)(8.1)

(4.1)(4.1)

(7.1)(7.1)

For ,

Similar to the Hermite differential equation, the general solution to the Laguerre equation is linear combination of Kummer functions.This particular Kummer function, has a special name, Laguerre function

which can be expressed in simple form when is integer.

In Maple, Laguerre polynomials are predefined as LaguerreL(n,x)The first few Laguerre polynomials are:

= = simplify

1

= = simplify

= = simplify

(8.1)(8.1)

(4.1)(4.1)

(7.1)(7.1)

Orthogonality

forms an orthonormal basis set for :