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VII. Note on the separation of a Fraction into Partial 

 Fractions. By I. J. Schwatt *.. 



THE following method for the separation of a fraction, 

 whose denominator is a power of a linear expression, 

 into partial fractions is simpler than the methods I have 

 given before f. 



To separate into partial fractions 



n 



2 m a x n ~ a 

 Let iv=y—a, then 



n 



% m a {y—a) n -°- 

 Wy-a)^^—^- ■ . . . (1) 



or 



F(.>j-a) = y-* 2 m."i"(-iy ("*")*"— "of. (2) 



a = /3 = \ H / 



Letting a+/3=7, (2) becomes 



Y(y-a) = 2 m a 2 (-l^fc^y^*-*^-. (3) 



a=0 y=a \/ **/ 



Nov/ since 



« >i n y 



2 S Aa, 7 = 2 X Aa, 7, . . . . (-1) 



a = y — a y=:0 a=0 



therefore 



F(y-a) = 2 y-*-v I (-l>-m.("i")*^", • (5J 



Y=0 a=0 \ » ' 



We shall now distinguish between the two cases n < p and 



(i.) n <p. 



"We may write (5) in the form 



*(-n-<::> 



F(?/-a)= S „P-w-y) ' • 1 (>)X 



* Communicated by the Author. 



t Quarterly Journal of Mathematics, No. 174, 1913 j Archil 

 Mathematik und Phi/silc, xxii. 1914. 



