IQ4 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



INTEGRAL PROPERTIES 



9.044 



f +1 P m (x)P n (x) dx = ottmn 

 J-i 



2 .. 



if m = n. 



9.045 



(m - n) (m + n + i) / P m (x)P n (x) dx 



9.046 



-.... + Pn-lPn) 



EXPANSIONS IN LEGENDRE FUNCTIONS 



9.050 Neumann's expansion: 



fln- ( + r i J(x}Pn 

 J-i 



9.051 Any polynomial in ^ may be expressed as a series of Legendre's poly 

 nomials. If /(#) is a polynomial of degree n: 



SPECIAL EXPANSIONS IN LEGENDRE FUNCTIONS 



9.060 For all positive real values of n : 



i + cos mr ( _, , ~\ <w? ^ . m 



i. cos nO = - 7-^ -^ P (cos ff) + ,* _ , P 2 (cos 0) 



