* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download PHYS 301 -- Introduction to Mathematical Physics Chapter 12 Series
Survey
Document related concepts
Renormalization group wikipedia , lookup
Two-body Dirac equations wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Inverse problem wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Numerical continuation wikipedia , lookup
Perturbation theory wikipedia , lookup
Mathematical descriptions of the electromagnetic field wikipedia , lookup
Least squares wikipedia , lookup
Routhian mechanics wikipedia , lookup
Computational fluid dynamics wikipedia , lookup
Transcript
PHYS 301 -- Introduction to Mathematical Physics Chapter 12 Series Solutions of Differential Equations…(Section 1-4 Legendre Polynomials) (i) For a second order linear ODE that is nonlinear with non-constant coefficients, we cannot use methods in Chapter 8 directly. However, if a solution of it can be expressed into a power series of x, it is possible to find it by substituting the unknown series into the ODE and solve for relationships between coefficients of different terms. (ii) As an example, we consider the Legendre differential equation, which is frequently encountered in solving partial differential equations using spherical coordinates. (iii) If l is a non-negative integral in the Legendre differential equation, one series solution of it only has terms up to x l , and so it is called Legendre polynomials Pl (x) , defined with Pl (1) = 1. (iv) Pl (x) can also be found using the Gram-Schmidt Method (pp 182-183) by requiring 1 1 € = 0 if l ≠ m . In fact, it can also be shown€ that ∫ x ∫ P (x)P (x)dx l m −1 m Pl (x)dx = € 0 if m < l . Also, −1 € Pl (x) is an even function with respect to x for even l, and is an odd function for odd l. (v) € € € 1 € dl 2 Pl (x) = l (x −1) l . Pl (x) can also be expressed in Rodrigues’ formula: € l 2 l! dx Key equations: € (12.2.1) Legendre differential equation. € (12.2.7) Series Solutions of the Legendre differential equation. (12.2.8) The first few Legendre Polynomials. (12.4.1) Legendre Polynomials expressed in Rodrigues’ formula. Examples: 12.1.2, 12.1.3, 12.2.2, 12.3.6, 12.4.4