Lecture 16: Special Functions. In Maple, Hermite polynomials are predefined as HermiteH(n,x) The first few Hermite polynomials are: simplify.
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1 Lecture 16: Special Functions 1. Key points Hermite differential equation: Legendre's differential equation: Bessel's differential equation: Modified Bessel differential equation: Spherical Bessel differential equation: Airy differential equation: Lagguerre differential equation: Maple HermiteH(n,) LegendreP(n,), LegendreQ(n,) BesselJ(n,), BesselY(n,) HankelH1(n,), HankelH2(n,) BesselI(n,), BesselK(n,) AiryAi(), AiryBi() LaguerreL(n,) Hermite equation Hermite differential equation When is integer, Hermite polynomials (2.1) (2.2) is a solution to (2.1). In Maple, Hermite polynomials are predefined as HermiteH(n,) The first few Hermite polynomials are: 1
2 Plots Recursive relation The Hermite polynomials Hermite equation. detemrined by the following recursive relation are solution to the (2.3) Orthogonality
3 (2.4) The second solution to the Hermite equation is the second kind Hermite function rarely used in physics, we don't discuss it here.. Since it is Using Maple to solve Hermite equation For n=2, The first term is the second kind Hermite function. Under typical boundary conditions in physics, the integral constant _C1 is zero. The second term is basically the Hermite polynomial. (2.5) General solution by Maple (2.6) Legendre equation Legendre's differential equation of degree n (0th order) Two linearly independent solutions to (3.1) is known as the first kind of Legendre polynomials and the second kind of Legendre function. (3.1) First kind Legendre polynomials (3.2) In Maple, Legendre polynomials are predefined as. = 1 = =
4 = Orthogonality forms an orthonormal basis set for. (3.3)
5 Recursive equation (3.4) Second kind Legendre function is not common in physics problem. Solution by Maple (3.5) General solution by Maple (3.6) General Legendre equation Legendre's differential equation where and are integers and. (mathematically speaking non-integer values are allowed but not popular in physics.) (4.1) Associate Legendre function (4.2)
6 (4.3) is a solution to the general Legendre differential equation. The second kind of assocate Legendre function 1 (4.4) (4.5) Solution by Maple
7 (4.6) (4.7) Bessel equation Bessel's differential equation (5.1) Two linearly independent solutions are known as Bessel function, and Weber function. Hence, a geneal solution is given by The second solution is also called Neuman function and denoted as. (5.2) General solution by Maple (5.3) Hankel functions independent solutions. are also a pair of linearly In Maple, these functions are predefined as BesselJ(n,), BesselY(n,), BesselH1(n,), and BesselH2 (n,). Since they can be epressed only by infinte series, Maple cannot epress them in closed forms.
8 (5.5) Plot
9 Bessel functions are not orthogonal! Modified Bessel equation Modified Bessel differential equation (6.1) Two linearly independent solutions are the first kind and second kind of modeified Bessel functions, and, respectively. General solution by Maple (6.2) They are related to the Bessel functions as follows:
10 (6.3) In Maple, the modified Bessel functions are predefined as BesselI(n,) and BesselK(n,) Plot Modified Bessel function of the first kind Modified Bessel function of the second kind
11 Spherical Bessel equation Spherical Bessel differential equation Two linearly independent solutions are spherical Bessel and spherical Nuemann functions,. (7.1) and They are related to the Bessel functions as follows:
12 (7.2) (7.3) Spherical Bessel functions can be epressed in simple form. For eample, and. combine symbolic combine 3 2 symbolic 2 Plot Spherical Bessel and.
13 Spherical Bessel and.
14 Airy equation Airy differential equation (8.1) Two linearly independent solutions are the first and second kind of Airy functions, Ai() and Bi(), respecctively. They are related to modified Bessel functions as follows:
15 (8.2) In Maple, the Airy functions are predefined as AiryAi() and AiryBi(). Plot 0 5
16 General Solutions by Maple Since (8.3), the second term is usually elimnaed by physical boundary condition. Lagguerre equation Lagguerre differential equation (9.1) Laguerre's polynomial is a solution. (9.2)
17 In Maple, Laguerre polynomials are predefined as LaguerreL(n,) The first few Laguerre polynomials are: 1 Plot Orthogonality forms an orthonormal basis set for :
18 (9.3) General Solutions by Maple (9.4) (9.5) The first term is since it diverges as.. The second term is usually elimnated by physical boundary conditions
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Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
d 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.
4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal