PHYS 404 Lecture 1: Legendre Functions

Transcription

1 PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS LECTURE 1 DR. V. LEMPESIS 1

2 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions of Legendre s differential equation that appear when we separate the variables of HelmholG equation, Laplace equation or Schrodinger equation using spherical coordinates. PHYS LECTURE 1 DR. V. LEMPESIS 2

3 Legendre Functions physical justification Almost all the cases are specific cases of the general differential equation: { 2 + k 2 U (r)}ψ ( r) = 0, k = const (1) Following the method of separation of variables, i.e. by considering a solution of the form: Ψ( r) = R( r)y ( θ,ϕ) (2) PHYS LECTURE 1 DR. V. LEMPESIS 3

4 Legendre Functions physical justification The differential equation (1) splits in two simpler equations ( 1 " sinθ θ sinθ % $ '+ 1 2 ) * # θ & sin 2 θ φ 2 +, - Y θ,ϕ ( ) = λy θ,ϕ ( ) (1.3) 1 r 2 d dr dr r2 dr + k 2 U (r) λ r 2 R = 0 (1.4) PHYS LECTURE 1 DR. V. LEMPESIS 4

5 Legendre Functions physical justification Eq. 3 is independent from function U(r) and is called angular equation. Solutions of this equation could be: trigonometric functions, Legendre functions, associated Legendre functions and the spherical harmonics. Eq. 4 does depend on function U(r) and is called radial equation. Solutions of this equation could be: Bessel functions, Hermite functions and Laguerre functions. PHYS LECTURE 1 DR. V. LEMPESIS 5

6 The angular equation Try to consider for Eq. 3 a solution of the separating variable form: Y ( θ,ϕ) = Θ( θ)φ( ϕ) (1.5) Then the radial equation is split into two equations: 1 sinθ d " dθ% " $ sinθ '+ λ µ 2 % $ 'Θ = 0 (1.6) dθ # dθ & # sin 2 θ & d 2 Φ d 2 φ + µ 2 Φ = 0 (1.7) PHYS LECTURE 1 DR. V. LEMPESIS 6

7 The associated Legendre equation If we make the substitutions x = cosθ, u(x) = Θ( θ), λ=ν(ν +1) (1.8) Then Eq. 1.7 takes the form d dx " $ # ( 1 ) du x2 % " '+ ν(ν +1) µ 2 $ dx & # 1 x 2 % 'u = 0 (1.9) & This is the so called associated Legendre equation Substitution x=cosθ is needed in order to give us a DE with rational coefficients for which we have efficient tools for its solution; method of power series PHYS LECTURE 1 DR. V. LEMPESIS 7

8 The associated Legendre equation The associated Legendre Equation has two linearly independent solutions: The associated Legendre function of first kind: The associated Legendre function of second ν kind: Q µ x ( ) In physical problems Eq. 1.7 has single valued and periodical solutions only if m =integer. In this case Φ m = exp( imφ ) / 2π (1.10) P µ ν ( x) PHYS LECTURE 1 DR. V. LEMPESIS 8

9 The associated Legendre equation Then in this case the associated Legendre Equation is wrizen as d dx " $ # ( 1 ) du x2 % " '+ $ ν(ν +1) m2 dx & # 1 x 2 % 'u = 0 (1.10) & PHYS LECTURE 1 DR. V. LEMPESIS 9

10 Legendre Diff. Equation If in the previous equation we consider m=0 and also consider that u(x) = y(x) m=0 then we get the Legendre differential equation: d dx " $ # ( 1 ) dy x2 % '+ν(ν +1) y = 0 (1.11) dx& Eq has finite solutions at the points where x = ±1 only if ν = n is a positive integer. PHYS LECTURE 1 DR. V. LEMPESIS 10

11 Legendre Functions ( ) Q ( n x) Legendre functions (or polynomials) P n x, are a solution of Legendre differential equation about the origin (x = 0). ( ) The polynomials Q l x are rarely used in physics problems so we are not going to deal with them further. On the contrary the polynomials P l x, which appear in many physical problems, may be defined by the so called generating function. ( ) PHYS LECTURE 1 DR. V. LEMPESIS 11

12 Legendre Polynomial The generating function The generating function of Legendre polynomials: (1.12) Which has an important application in electric multipole expansions. If we expand this function as a binomial series if we obtain t <1 (1.13) PHYS LECTURE 1 DR. V. LEMPESIS 12

13 Legendre Polynomial The generating function Using the binomial theorem we can expand the generating function as follows: g(t, x) = ( 1 2xt + t 2 ) 1/2 = ( 2n)! ( ) n (1.14) n=0 2 2n n! ( ) 2 2xt t 2 From the above series we can get the values of the Legendre functions. PHYS LECTURE 1 DR. V. LEMPESIS 13

14 Legendre Polynomial plots PHYS LECTURE 1 DR. V. LEMPESIS 14

15 The Recurrence Relations Legendre polynomials obey the following recurrence relations: 2n +1 ( ) xp n (x) = ( n +1) P n+1 (x) + np n 1 (x) ' P n+1 ' P n+1 ' (x) + P n 1 (x) = 2xP ' n (x) + P n (x) ' (x) P n 1 (x) = ( 2n +1) P n (x) ' P n+1 (x) = ( n +1) P n (x) + xp ' n (x) ' P n 1 (x) = np n (x) + xp ' n (x) ( 1 x 2 ) P ' n (x) = np n 1 (x) nxp n (x) These relations are valid for (1.15) PHYS LECTURE 1 DR. V. LEMPESIS 15

16 Legendre Polynomials Special Properties Some special values (1.6a) and (1.16b) Where (called double factorial) PHYS LECTURE 1 DR. V. LEMPESIS 16

17 Legendre Polynomials Special Properties The Parity property:(with respect to ) (1.17a) (1.17b) If n is odd the parity of the polynomial is odd, but if it is even the parity of the polynomial is even. Upper and lower Bounds for (1.18) PHYS LECTURE 1 DR. V. LEMPESIS 17

18 Legendre Polynomials Orthogonality Legendre s equation is a self-adjoint equation, which satisfies Sturm-Liouiville theory, where the solutions are expected to be orthogonal to satisfying certain boundary conditions. Legendre polynomials are a set of orthogonal functions on (-1,1). (1.19) where PHYS LECTURE 1 DR. V. LEMPESIS 18

19 Legendre Polynomials Legendre Series According to Sturm-Liouville theory that polynomial form a complete set. Legendre (1.20) The coefficients are obtained by multiplying the series by and integrating in the interval [-1,1] a n = 2n f (x)p n (x) (1.21) PHYS LECTURE 1 DR. V. LEMPESIS 19

20 Alternate definitions of Legendre polynomials Legendre polynomials can be defined with the help of the so called Rodriguez formula: P n (x) = 1! # 2 n n! " d dx (1.22) The Rodrigues formula provides a means of developing a means of developing an integral representation in the complex plane with the Schlaefli integral : P n (z) = 1! d $ # & 2 n n! " dz % n $ & % ( z 2 1) n = 2 n 2πi n ( x 2 1) n ( t 2 1) n! dt Contour ( t z) n+1 encloses t=z. PHYS LECTURE 1 DR. V. LEMPESIS 20