96 Mr Orr, On the product J m (x) J n {x). 



Making use of the equation* connecting 

 at+ n F($+n; l + 2n; 2x), 

 x i- n F{\-n; l-2n; 2x), 



and <}>(% + n, l - n ; - —J , 



we obtain the result 



I-n - I n = \j — smnire- x (j> U + n, \-n; - ^j . . .(14), 



wherein the argument of every power x m lies between — Trur and 



And by writing x = ye~ ni in this we have 

 /_„ (y) + I n (y)-i cot mr [I_ n (y) - I n (y)} 



=\Z^ ev K i+n ^- n 'k) (15) - 



wherein the argument of every power y m lies between and 

 2m7r. 



If the modulus of y be large and its argument lies between 

 and 7r/2, (14) shows that the left-hand member is approximately 

 the same as I_ n (y) + I n (y) (even when n is integral). An equa- 

 tion analogous to (15) may be deduced from (15) by changing the 

 sign of i, and holds provided the argument of every power y m lies 

 between and — 2ra7r. 



Thus, provided y is large and its real part positive, excluding 

 zero however, we have the approximate equality 



l-n(y) + in{y)h{J^jey (is a), 



reducing when n is an integer to 



I n {y)h{2iry)-Uy (156). 



If we define K n (x) by means of the relation 



K n (x) = ' 7 r I - n ^}~ In ^\ 



z sin nir 



* See equation (26) of the paper referred to above. 



