Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics University of California, Santa Cruz October 2010 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions There are several special functions that recur in many branches of physics. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. The spherical harmonics, more generally, are important in problems with spherical symmetry. They occur in electricity and magnetism. They are important also in astrophysics and cosmology, where they play the role of sines and cosines in fourier expanding functions on the sky. Legendre polynomials and legendre functions more generally solve the θ equations. Bessel functions arise in problems with spherical symmetry, but actually occur also more broadly. In quantum mechanics, particular instances solve the free particle radial equation in spherical coordinates, and again in cosmology, they appear as solutions to a number of problems. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Start with Laplaces’s eqn. in spherical coordinates: 1 @2 1 @ @Φ 1 @2Φ (rΦ) + (sin θ ) + = 0: (1) r @r 2 r 2 sin θ @θ @θ r 2 sin2 θ @φ2 Separate variables: u(r) Φ = P(θ)Q(φ): (2) r Leads to Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions 1 d 2u 1 d dP 1 d 2Q r 2 sin2 θ + (sin θ ) + = 0: (3) u dr 2 r 2 sin θP dθ dθ Q sφ2 The last term must be a constant: d 2Q = −m2Q (4) dφ2 or Q = e±imφ: (5) Singlevaluedness ) m integer. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Now want equations for P, u. Divide by sin2 θ; 1 d 2u 1 d dP m2 r 2 + (sin θ ) − = 0; (6) u dr 2 sin θP dθ dθ sin2 θ Second and third terms are independent of r, so can again introduce separation constant: d 2u u − `(` + 1) = 0 (7) dr 2 r 2 and 1 d dP m2 (sin θ ) + `(` + 1) − P = 0: (8) sin θ dθ dθ sin2 θ Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions u eqn. has solutions r a, with a(a − 1) = `(` + 1). So u = ar `+1 + Br −`: (9) If solving equation in all of space, can reject solution singular at the origin. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions For the equation for P, we first write d 1 d x = cos θ; = − (10) dθ sin θ d cos θ so the equation can be rewritten as: d dP m2 (1 − x2) + `(` + 1) − P = 0: (11) dx dx 1 − x2 The case m = 0 is known as the ordinary Legendre differential equation; the case of non-zero m is known as Legendre’s equation. The solutions of the first are known as Legendre polynomials; of the second as associated Legendre functions. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Solution by series method Starting with the ordinary Legendre equation: d dP (1 − x2) + `(` + 1)P = 0: (12) dx dx try a solution: 1 α X j P(x) = x aj x (13) j=0 Substituting and equating powers of xj , gives the relation: (α + j)(α + j + 1) − `(` + 1) a = a : (14) j+2 (α + j + 2)(α + j + 1) j Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions In addition, either a0 = 0, or α(α − 1) = 0, or a1 = 0 or (α + 1)α = 0. As a result, the series consists either of even terms, starting with x0, or odd terms starting with x1. For j large, a j+2 ! 1; (15) aj 1 this is the same as the series for 1−x2 , and so diverges as x ! 1. In order that the series converge over the whole angular range, we require that the series terminates, which occurs if ` is a positive integer. Exercise: Verify the recursion relation above. With the convention that P(1) = 1, determine the first three Legendre polynomials. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Properties of the Legendre Polynomials ` P`(x) = (−1) P`(x): (16) Like eik·x in Fourier series, a complete, orthogonal set. Orthogonality: Multiply legendre’s equation by P`0 and integrate over x: Z 0 d 2 dP` dx P 0 (1 − x ) + `(` + 1)P = 0: (17) ` dx dx ` Integrate by parts; then subtract the same equation with ` $ `0. The surface term vanishes since 1 − x2 = 0. Then, for ` 6= `0, Z 1 dxP`(x)P`0 (x) = 0: (18) −1 We will do the case ` = `0 (normalization) shortly. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Generating Function for the Legendre Polynomials Just what is says: a way of generating the Legendre functions explicitly, without directly solving the differential equation. Start with the fact that 1 G(~x; ~x 0) = (19) j~x − ~x 0j solves the laplace equation, the angular part of which (ignoring φ) is Legendre’s equation. For r 0 < r, expand: 1 1 1 1 = = (20) j~x − ~x 0j (r 2 + r 02 − 2rr 0 cos θ)1=2 r r 02 r 0 1=2 (1 + r 2 − 2 r cos θ) 1 1 X r 0 = ( )`A (cos θ): r r ` `=0 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Note that the first term is the function u(r). Claim is that A` = P`. Note for starters, that if A` satisfies the the condition A`(1) = 1: The differential equation, in fact, follows by substituting the expansion. For any fixed `, because u` satisfies our previous equation, A` must satisfy the P` equation. The expression, indeed, must hold for all θ and r, so it must hold term by term. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions We can summarize: 1 1 X ` g(t; x) = = P`(x)t : (21) (1 − 2xt + t2)1=2 `=0 We can extract explicit form for the P`’s by extracting coefficient of t`: E.g. P0 = 1; P1 = x (22) We can extract explicit properties of P`’s by manipulating g(t; x). Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Normalization integral: Z 1 Z 1 1 1 1 X X ` `0 dx = ( )( )P (x)P 0 (x)t t : (23) 1 − 2tx + t2 ` ` −1 −1 `=0 `0=0 Right hand side is Z 1 X 2 2` P`(x) t : (24) −1 Left hand side is elementary: 1 1 + t ln( ): (25) t 1 − t Expand in powers of t, using 1 X (−1)n+1 log(1 + a) = an: (26) n n=1 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions So we have 1 1 + t 2 t2n+1 log = : (27) t 1 − t t 2n + 1 So Z 1 2 2 dxPn dx = : (28) −1 2n + 1 Exercise: Differentiate g with respect to t and derive the recursion relation: (2n + 1)xPn − (n + 1)Pn+1 − nPn−1 = 0: (29) Similarly, differentiate with respect to x: dP dP `+1 − `−1 − (2` + 1)P = 0: (30) dx dx ` Other recursion formulas, integral formulas, can be derived similarly (see, e.g., Arfken and Weber). Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions With these ingredients, we can expand any function of cos(θ) (i.e. functions which are single-valued and otherwise well behaved as functions of θ) in terms of the Legendre polynomials. X f (cos θ) = a`P`(cos θ) (31) with r 2` + 1 Z 1 a` = f (x)P`(x): (32) 2 −1 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Associated Legendre Functions Recall the more general differential equation: m2 (1 − x2)v 00 − 2xv 0 + `(` + 1) − v = 0: (33) 1 − x2 This equation is solved by the associated Legendre functions: d m Pm(x) = (−1)m(1 − x2)m=2 P (x): (34) ` dxm ` The constant in front is conventional. That this solves the equation is shown by repeatedly differentiating Legendre’s equation. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Exercise: Prove that the functions in eqn. 34 solve the full equation for positive m by repeated differentiation. For negative m, argue that the same is true by arguing that −m(x) m P` / P` (x): (35) Not exercise: The normalization integral for the associated Legendre functions is: Z 1 m 2 (` + m)! P`0 (x)P` (x)dx = δ``0 : (36) −1 2` + 1 (` − m)! Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Spherical Harmonics Solve the full θ, φ equations. Convention is to normalize to unity when integrated over the sphere. s 2` + 1 (` − m)! Y (θ; φ) = Pm(cos θ)eimφ: (37) `m 4π (` + m)! ` Very useful is the addition theorem: 1 ` ! 1 X X 1 r ` = 4π < Y ∗ (θ0; φ0)Y (θ; φ): j~x − ~x0j 2` + 1 `+1 `m `m `=0 m=−` r> (38) Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Bessel Functions These arise frequently in problems with cylindrical symmetry.