Lecture 19: Ordinary Differential Equations: Special Functions Key points Hermite differential equation: Legendre's differential equation: Bessel's differential equation: Modified Bessel differential equation: Spherical Bessel differential equation: Airy differential equation: Laguerre differential equation: Maple HermiteH(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 called confluent hypergeometric functions. In Maple, they are predifined functions, and. (2.1) The two independent solutions for the Hermite differential equation is and
For integer, Hermite polynomial. is a solution to the Hermite differential equation.. For, (2.3) In Maple, Hermite polynomials are predefined as HermiteH(n,x) The first few Hermite polynomials are: 1
The Hermite polynomials Hermite equation. determined by the following recursive relation are solution to the 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)
General Solution (3.1) For, (3.2) 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 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. 1 Solution by Maple
(4.1) (4.2) Bessel equation Bessel's differential equation General solution by Maple (5.1) 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 independent solutions. Even for integer, there is no simplex expression: For, are also a pair of linearly (5.2) 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.)
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. (6.1) In Maple, the modified Bessel functions are predefined as BesselI(n,x) and BesselK(n,x). There is no
simple expression of the modified Bessel functions even for integer. (6.2) 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
(7.1) 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, (7.2) For general integer, Spherical Bessel functions can be expressed in simple form. For example, symbolic x symbolic x 2
symbolic symbolic Note that the spherical Neumann functions diverge at. Spherical Bessel Spherical Neumann function Airy equation Airy differential equation General Solution
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: (8.1) 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. (9.1)
For, (9.2) 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: 1
Orthogonality forms an orthonormal basis set for :