Chapter4.ver5c

Chapter4.ver5c - Chapter 4 Boundary Value Problems in...

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Chapter 4 Boundary Value Problems in Spherical and Cylindrical Coordinates 4.1 Laplace’s Equation in spherical coordinates Laplace’s equation in spherical coordinates, ( r, θ, φ ) , has the form 1 r 2 ∂r · r 2 Φ ( r, θ, φ ) ¸ 1 r 2 L · L Φ ( r, θ, φ )=0 . (4.1) where L · L 1 sin θ ∂θ · sin θ Φ ( r, θ, φ ) ¸ + 1 sin 2 θ 2 ∂φ 2 (4.2) Multiplying Eq. 4.1 by r 2 , dividing by Φ ( r, θ, φ )= R ( r ) Y ( θ,φ ) and rewriting, 1 R · r 2 R ( r ) ¸ = 1 Y L · L Y ( ) . (4.3) This equation must be satis f ed for all ( r, θ, φ ) As r, θ, φ are independent variables, Eq. 4.3 can only be satis f ed if both sides equal a constant which we choose this constant to be c ( c +1) . Then, L · L Y ( c ( c Y ( ) (4.4) and d dr · r 2 d dr R ( r ) ¸ c ( c R ( r (4.5) Eq. 4.4 can be further separated using Y ( V ( θ ) W ( φ ) : 1 V ( θ ) sin θ · sin θ V ( θ ) ¸ + 1 V ( θ ) sin 2 θc ( c V ( θ 1 W 2 2 W (4.6) Again, since θ , φ are independent variables both sides must equal a constant (the second separation constant) which we take to be m 2 2 2 W ( φ m 2 W ( φ ) or (4.7a) W m ( φ A m e imφ + B m e imφ (4.7b) and sin θ · sin θ V ( θ ) ¸ +[sin 2 ( c m 2 ] V ( θ (4.8) 73
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Substituting x =cos θ Eq. 4.8 becomes the associated Legendre equation : d dx · (1 x 2 ) d dx F ( x ) ¸ +[ c ( c +1) m 2 1 x 2 ] F ( x )=0 . ; x θ (4.9) with two linearly independent solutions called the associated Legendre functions of the f rst, P | m | c ( x ) , and second kind, Q | m | c ( x ) . F c, | m | ( x )= A c, | m | P | m | c ( x )+ B c, | m | Q | m | c ( x ) (4.10) One can show that:the P | m | c (cos θ ) canbemadetoconvergeforall 0 θ π if : 0 <m = | m | cc =0 , 1 , 2 . == >P | m | c (cos θ ) converge .. The Q | m | c (cos θ ) ,howeverd ivergea t cos θ = 1 for the above conditions on m and c .T h e Q | m | c (cos θ ) are possible solutions when | cos θ | < 1 . In our problem we expect the functions to be well behaved for all | x | = | cos θ | 1 . A special case is found when | m | and these solutions, P 0 c ( x P c ( x ) are called Legende functions or Legendre polynomials. The P 0 c ( x ) are normalized so that: P 0 ( x )=1 (4.11) P 1 ( x x P 2 ( x 1 2 ¡ 3 x 2 1 ¢ P 3 ( x x 2 ¡ 5 x 2 3 ¢ P 4 ( x 1 8 ¡ 35 x 4 30 x 2 +3 ¢ The associated Legendre polynomials , P | m | c ( x ) P m c ( x ) , of degree c and order m. are related to the Legendre polynomials by P m c ( x ¡ 1 x 2 ¢ m/ 2 d m P c ( x ) dx m ,m 0; (4.12) P m c ( x )=( 1) m P | m | c ( x ) ,m< 0 This particular sign convention is just one possibility. Clearly these are only non-zero when c | m | . F ina l lyweno tetha ta t P 0 c (1) = 1 (by de f nition), P 0 c ( 1) = ( 1) c , and, if m 6 ,P m c ( ± 1) = 0 . The general symmetry of these functions for x →− x is P m c ( x 1) c + m P m c ( x ) Legendre’s functions of the second kind Q c ( x ) , have singularities at x = ± 1 . This solution would be one of the set known as Legendre functions of the second kind. The f rst two Legendre functions of the second kind are, for | x | 1 , 74
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Section 4.2 Spherical harmonics Q 0 ( x )= 1 2 ln μ 1+ x 1 x =tanh 1 ( x ) (4.13) Q 1 ( x x 2 ln μ x 1 x 1 We note the logarithmic divergences for Q n ( x ) at x = ± 1 .
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Chapter4.ver5c - Chapter 4 Boundary Value Problems in...

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