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XY. Note on the Partial Fraction Problem. 

 By I. J. SCHWATT *. 



TO the methods I have given before t I add the following : 

 I. To separate 



n 



X m a x n ~ a 

 F(x?) = -^ , where n< 2p, 



into partial fractions. 



FW ,iAg£ CD 



j8=l # + C 



This relation may be written 



F W = ^±B« + J^l .... (2) 

 where Jt _ 1 



y=l 7 V = K + 1 / 



From (2) follows 



n 



S m ft ^-= (A^+B,)^W +(^ + (5tW- • (3) 



a=0 



Substituting in this identity ic K for x, we have 



Sm a cr a ^- a =(/> K (^)(AA + B K ) ... (4) 



a=0 



To obtain A K and B K we separate the first member of (4) 

 into its real and imaginary parts. 

 We have 



n n 



X m a c n ~ a i n - a = 2 m n _ a 4i a = m n — m n _ 2 c2 + m„_ 4 ^— 



a=0 a=0 



, / in DO 2 0,.r 3 



re- 1 ] ■Pr 1 i«i 



+ m n . 5 c 5 -.. .. + (-1) m n _r^n_ lC/c J, 



or L 2 J 



GO PirG 



2 m^-*— = 2 (-l) a m n _ 2 /; + i 2 (-l)X- 2a -i^ +1 - (5) 



a=0 a=0 !a=0 



* Communicated by the Author. 



f Quarterly Journal of Mathematics, no. 174 (1913) ; Archiv der 

 Mathematik und Physik, xxii. 1914 ; Phil. Mag. vol. xxix., January 1915, 



