1871.] Linear Differential Equations, 283 



Differentiating and putting x—0, 



iz . diz 



_^ (iz—a) K + i (iz—^+ 1 {iz—y)"+^ 

 d . iz 



;Aa + ]J 



B =r-r- 



)*(is-/3)w +1 (/z— y)" +1 



and so we may proceed. The integral is thus completely determined. 

 A similar process will give us the more general integral 



1 



(iz—a) K+1 {iz—l3y+ ] (iz—yy+i («— £) w+1 " 



where a is essentially negative, and ft, y . . . . £ essentially positive*. 



The same process will hold good when p, v , . . . are fractional ; for it is 

 manifest that the integral 



c ixz /Hi? 



=Pe«*-fQe^+ReV*; 



x 



(h-ay+ 1 (iz-P)*+ i (iz-Yy +1 



where, however, 



q= ... +~St + • • • +a^- 2 +a^-+a a 



with a similar expression for R. 



Now, as x increases without limit, the series 



Ay _|_ Ay-f-l , 



converges; for since x is arbitrary, the ratio of two consecutive terms, ^ v ~ k+l 3 



may be made as small as we please, however great v becomes. Hence, as x 



increases without limit, the series J V Z* + . . . approaches zero. Therefore, 



as we suppose /3 positive, QeP* will increase without limit ; and therefore 

 as Q is supposed multiplied by an arbitrary constant, we must have Q=0. 

 Hence the value of the integral where p, v, . . . are supposed fractional will 

 also be Ye ax , a being negative, and the constants in P determined as before. 

 Next, consider the integral 



,co e**(z-o(,y-h?z 



(z—$y+\z-Yy+ i> 



where x is supposed negative, a and /3 essentially positive, /3 less than 



* It is proper here to remind the reader that the integral 



f 30 = 2 7 r/3 m - 1 e-^ 

 „ _ w (a F (") 

 does not hold good when a is negative. 



z 2 



