24 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



1.850 Definite Sums. From the indefinite sum, 



S/0) = F(x) + C, 

 a definite sum is obtained by subtraction, 



a+nh 



.f(x) = F(a + nh) F(a + mh). 



a+mh 



1.851 



a+nh 



* + h)+f(a+2h) + +f(a + fT 



By means of this formula many finite sums may be evaluated. 

 1.852 



a+nh 



/j(x b)(x b h)(x b 2h) . . . . (x b k ] 



a 



= (a - b + nh) (a - b + n - ih) . . . . (a - b + n - kti) 



(k + i)h 



(a - b)(a -b-h) . . . . (a-b-kh) 

 (k + i)h 



1.853 



n(n - i)(n - 2) .... (n - k) Jk 

 (k+i) 



1.854 If f(x) is a polynomial of degree m it can be expressed : 



/(*) = AQ + Ai(x - a) + ^4 2 (* - )(* - a - h) + . . . . 

 + A m (x - a) (x - a - h) , . (x - a - m - ih), 



a+nh 



1.855 If f(x) is a polynomial of degree (m i) or lower, it can be expressed 



f(x) = AQ 4- A\(x + mh) + A z (x + mh) (x + m ih) 



+ .... + A^^x + mh) . . . (x + 2/0 

 and, 



c+nA 



mh \ a(a + h) . . . (a + m - iti) 



