174 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



8.242 If F(D) is a polynomial in D, 



1. F(D)e mx = e mx F(m). 



2. F(D)e mx X = e mx F(D + m)X. 



3. e mx F(D)X = F(D - m)e mx X. 



8.243 If F(B) is a polynomial in 6, 



1. F(6)x m = x m F(m). 



2. F(d)x m X = x m F(d + m)X. 



3. x m F(6)X = F(B - m)x m X. 



8.244 x"~ =6(6-1) (6 -2) (6 - m f i). 



INTEGRATION IN SERIES 



8.250 If a linear differential equation can be expressed in the symbolic form 



DrF(0)+/(0)];y=o, 

 where F(6) and/(0) are polynomials in 6, the substitution, 



_ 



leads to the equations, 



flo/(p) = o, 



+aif(p + m) = o, 

 aiF(p + m) + (hf(p + 2m) = o, 

 (hF(p + 2m) + a s f(p + 3/w) = o. 



8.251 The equation 



f(p) = o, 



is the " indicial equation." If it is satisfied a may be chosen arbitrarily, and the 

 other coefficients are then determined. 



8.252 An equation: 



may be reduced to the form 8.250, where, 



f(B) = 0(0 - m) 6(6 - i) (0 - 2) ..... (0-m + i). 



If the degree of the polynomial / is greater than that of F the series always con- 

 verges; if the degree of / is less than that of F the series always diverges. 



