Lecture 4: Series expansions with Special functions

Transcription

1 Lecture 4: Series expansions with Special functions 1. Key points 1. Legedre polynomials (Legendre functions) 2. Hermite polynomials(hermite functions) 3. Laguerre polynomials (Laguerre functions) Other kinds of special functions such as Bessel functions will be discussed in a later lecture of ordinary differential equations. Maple commands LegendreP HermiteH LagauerrL 2. Expansion with orthogonal functions Consider functions, defined in a region. They are said to be orthogonal if, where is a weight function which depend on the choice of. A function defined in the same region can be expressed as a linear combination of : where., and N is the dimension of the function space. (We will discuss this more in the linear algebra section. ) 3. Legendre polynomials Legendre function is a solution to the Legendre differential equation When is 0 or positive integer, is a polynomial of order defined on. (1)

2 1 x Orthogonality (2) For,

3 (3) Legendre series Functions defined on can be expanded in Legendre series where (4) Example Expand in Legendre series. Using the Heavicide function,. Common sense Heaviside function In Maple, the heaviside function is specified as. Legendre functions of even order have even parity. Since is an odd function, the coefficients of even order must be zero. Therefore, it is enough to consider only odd order terms. Common sense For any even function,. For any odd function,.

4 (5) (6)

5 Generating Function More detailed properties of Legendre polinomials: See mathworld Examples in Physics Dipole moment

6 A d R q Q 1 r Q 2 Find the potential at position A. If is small enough. This equivalent to say that two charges are at the same location. WE want to taken into account the distance d. Let and in the generating function, (7)

7 .. Coulomb interaction Consider two charges and separated by a distance. There interaction energy is given by This is mathematically complicated if cartesian coordinates are used. There is a convenient expansion using polar cordinates between and., and angle, where if and if. 4. Hermite polynomials Hermite polynomials, is a solution to Hermite differential equation When is 0 or positive integer, are polynomials of order. 1 (8)

8 Orthogonality. For,. Basis functions and weights There are various ways to define basis functions. 1., 2.,. Which one to use depends on the function to be expaned.

9 Hermite series Functions defined on can be expanded in Hermite series. where. Example Expand in Hermite series. Since and 0, cannot express. We use as basis function. Since Hermite polynomials of odd order has odd parity, the coefficints of odd order is zero. The terms of order higher than 8 are also all zero. Therefore, we need to consider only five terms (the orders of 0, 2, 4, 6, 8). (9) (10)

10 This plot shows that the expansion exactly matches to. Hermite expansion For a function and, the previous expansion does not work since diverges at. In physics, the following expansion is more popular. where. The expansion coefficient can be obtained by Note that. Example Expand with Hermite polynomials

11 Sinf the function is even, the odd terms vanish. Let us calculate the coefficients up to oder 10. Here erf() is the error function defined by The numerical value of the coeffcients are:. (11)

12 (12) RNow, we have an approximate expression of using Hermite polynimials. (13)

13 Agreement is not bad. Adding more terms will improve the agrement. Example in Physics: Quantum Harmonic Oscillator Schrödinger Equation: Let, and. Then, the Schrödinger equation becomes which is Hermite equation. Hence, and where. 5. Laguerre polynomials Laguerre function, is a solution to the Laguerre differential equation. When is 0 or a positive integer, is a polynomial of order. 1

14 Orthogonality. Laguerre series Functions defined on can be expanded in Laguerre series, where. Exercise

15 Expand original function. in the Laguerre series. Shows the first 5 terms and compare it with the (14) (15)

16 The approximation with the first 5 terms is not so bad. Homework: Due 9/12, 11am 3.1 Legendre polynomials Express as a linear combination of Legendre polynomials. 3.2 Hermite polynomials Express as a linear combination of Hermite polynomials.