IX. DIFFERENTIAL EQUATIONS 

 (continued} 



9.00 Legendre's Equation: 



, ON d?y dy 



(l ~ ^ dtf ~ 2X fa + n ( n + l)y = 



9.001 If ft is a positive integer one solution is the Legendre polynomial, or 

 Zonal Harmonic, P n (x): 



P M _ ( 2 ^ ! / x n _ n ( n ~ i) a.,-2 , n(n,-i)(n-2)(n- 3 ) \ 



X ~ 2 ( 2 n-i) X ' 2-4-(2- ' ' 



9.002 If n is even the last term in the finite series in the brackets is: 



9.003 If n is odd the last term in the brackets is 



9.010 If n is a positive integer a second solution of Legendre's Equation is the 

 infinite series: 



2(2^+3) 



(ft +i)O 



(n+3) 



, 



9.011 



P 2n (cos 0) - (-i)-M- 2 1 sin- tf - sin cos^ 



9.012 



(cos-(-i)"^<sin-0cos0- 



COS 



- 2) 2 .... 



(Brodetsky: Mess, of Math. 42, p. 65, 1912) 

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