LegendreGeneratingFunction

# LegendreGeneratingFunction - Legendre Generating Function...

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1 Legendre Generating Function. Ian R. Gatland, Georgia Institute of Technology. Physics 3122 Notes, June 17, 2010. The electrostatic potential is V ( r ) = 1 4 πε 0 ρ ( ʹ r ) d ʹ τ | r ʹ r | (1) and | r − ʹ r | 1 = [ r 2 2 r ʹ r cos θ + ʹ r 2 ] 1/ 2 = r 1 [1 2( ʹ r / r )cos + ( ʹ r / r ) 2 ] 1/ 2 (2) So, consider the Legendre generating function F ( x , h ) = (1 2 xh + h 2 ) 1/ 2 (3) and its expansion in powers of h F ( x , h ) = P ( x ) h = 0 (4) where the coefficients (the P ’s) are functions of x . To determine the differential equation satisfied by the P ’s we note that F / x = h (1 2 xh + h 2 ) 3/2 = ( h 2 xh 2 + h 3 )(1 2 xh + h 2 ) 5/2 (5) 2 F / x 2 = 3 h 2 (1 2 xh + h 2 ) 5/2 (6) ( hF ) / h = (1 2 xh + h 2 ) 1/2 + h ( x h )(1 2 xh + h 2 ) 3 /2 = (1 2 xh + h 2 + xh h 2 )(1 2 xh + h 2 ) 3/2 = (1 xh )(1 2 xh + h 2 ) 3/2 (7) and 2 ( hF ) / h 2 = x (1

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LegendreGeneratingFunction - Legendre Generating Function...

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