Chapter 3 - Chapter 3: Boundary-Value Problems in...

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() 2 0, The solutions are: ( ) ( ) ( ) ( ) : Legendre function of the first kind ( ) : Legendre funct 1 1 (3.9) 11 dd u dx dx ux AP x BQ x Px Qx x x u νν ν ⎡⎤ = ⎢⎥ ⎣⎦ =+ −≤ −+ + Legendre Equation : ion of the second kind : Gradshteyn & Ryzhik, "Table of Integrals, Series, and Products," Chs. 7 & 8. : Abramowitz & Stegun, "Handbook of Mathematical Ref. 1 Ref. 2 Functions," Ch. 8. 3.2 Legendre Equation and Legendre Polynomials Chapter 3: Boundary-Value Problems in Electrostatics: II We begin this chapter with 3 sections (Secs. 3.2, 3.5, & 3.6) on mathematics.
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3.2 Legendre Equation and Legendre Polynomials ( continued ) Rewrite the solution: ( ) ( ) ( ) ( ) diverges as 1. Hence, ( ) is not commonly used in physics. ( ) is finite for 1 and 1, but ( 1) diverges unless is an integer ux AP x BQ x Qx x Px x x P νν ν = + →± <= 1 (see p.105.) In many physics problems, boundary conditions require to be an integer. Since the form of the Legendre equation is unchanged if 1, we have ( ) ( ). Hence, when is P x −− →− − = an integer (denoted by ), negative is redundant. Thus, 0, 1, 2 and ( ) becomes a polynomial (properties on following pages). : The range ( is often encountered in 1 1) considered here l ll l P x Note x = −≤ " physics problems. Mathmatically, the range of ( ) and ( ) can be extended to the entire complex plane. Furthermore, can also be a complex number (See Gradshteyn & Ryzhik). Q x xi y +
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(1) l Q →∞ (1 ) (1 ) l l P −= 3.2 Legendre Equation and Legendre Polynomials ( continued ) Lengendre polynomials P 2 ( x ) - P 5 ( x ) [ P 0 ( x ) = 1, P 1 ( x ) = x ] Second Lengendre functions Q 0 ( x ), Q 1 ( x ), and Q 2 ( x ) 2 1 2! ( ) 1 , 0, 1, 2. .. (3.16) () l l ll l d ld x Px x l =− = Legendre Polynomial : 1 l P = ( 1 ) ( ) l P xP x
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3.5 Associated Legendre Functions and the Spherical Harmonics () 2 2 2 1 0 , for 1 1 T h e s o l u t i o n s a r e : : associated Legendre function of the first kind : asso 11 mm m m m x d d dx dx xA P x B Q x P Q x u u xu νν ν ⎡⎤ =− ⎢⎥ ⎣⎦ =+ −+ + Associated Legendre Equation : ciated Legendre function of the second kind (Refs.: Gradshteyn & Ryzhik; Abramowitz & Stegun) 1 1 2 21 The set ( ) is orthogonal: ( ) ( ) (3.21) It is also complete in index . Hence, any function ( ) can be expanded as ( ) ( ) ll l l l l Px P xPxd x lf x fx A δ ′′ + = = 0 (3.23) l =
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3.5 Associated Legendre Functions and the Spherical Harmonics ( continued ) Rewrite the solution: ( ) ( ) ( ) ( ) diverges as 1, hence is not commonly used in physics. ( ) is finite on the interval 1 1 only when is zero or a positive mm m m xA P x B Q x Qx x Px u x νν ν =+ →± −≤ () () () 2 (1 ) 22 2! integer ( 0, 1, 2. ..) and [p. 107.] , 1 , , 1, 0, 1, , 1 , Under these conditions, we have (for positive or negative ) ( 1 ) ( 1 ) m m l lm ml l l d dx l l l l m x x + == =− − − …… 1 1 0 (3.50) ( 1 ) ( ) ! with the properties: ( ) ( 1) ( ) (3.51) ! 2( ) ! ( 3 . 5 2 ) 21 ( ) ! m m ll mmm l l P x P x d x m P x δ + ′′ −= + + = +− =
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ϕ θ r x x y z s The set ( ) i complete in index in the sense any function ( ) : a fixed integer can be expanded as ( ) ( ) See (A.3) in Appendix A.
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This note was uploaded on 05/14/2010 for the course EAD 234 taught by Professor Ncl during the Spring '10 term at École Normale Supérieure.

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Chapter 3 - Chapter 3: Boundary-Value Problems in...

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