134 PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



C'" 



and therefore, m not being small, the important part of ' e^dt is ^if li ( ft)~ >h ; 



fm 

 d*t dt is J/8" 1 ; hence, the important part of 



f e^Wn+fdt is ^w n ir ! -( /8)~''-, provided iv' n is of lower order than w n fi"- with corre- 



Jo 



spending conditions for w" n , &c. Therefore, when /3~ v - is of an order higher than 

 unity, the most important part of S OT is ^w n rr ! *( /S)~ 1/2 , in this case the sum of an 

 order higher than that of the terms that compose it. 



The third case is when is small and u n + t contains an exponential factor of the 

 form c at+l>t ' ; as in the immediately preceding case, the result depends on the value of 



fm 

 e at + ltt* u ^ t fa 

 



If the real part of /3 is negative, or if a and ft are both pure imaginaries, the 

 important part of this integral is w n e at+pt 'dt, with the same restrictions as in the 



Jo 



previous case. 



The fourth case is that when /3=0, a. is small, aiid n n + t contains an exponential 



Jm 

 c at+ y (! w n+t dt, 

 o 



and, if the real part of j is negative, or if both and y are pure imaginaries, the 



~~r. 



important part of this integral is ' c^^^dt, with restrictions similar to those of 



. ii 



the two preceding cases. The integrals in the third and fourth cases have been fully 

 investigated and tabulated. 



2. Approximate Expressions for tlie Besse! Functions. 



Most of the expressions to be investigated in what follows have been given by 

 L. LORKNZ,* who obtained them from expressions for the sum of the squares of the 

 two solutions and the product of the two solutions. 



The investigation that follows derives them directly from the fundamental 

 expression for a solution of BESSEL'S equation and the passage -from the periodic form 

 of the expressions to the form involving real exponentials, which is insufficiently 

 treated by LORENZ, is traced. Writing 



^J s+Va (z) = 2V-% n) (-)W_ B _ Va (z) = 2V-'<x, 

 and using ScHLAFLi'sf formula for J n , it follows that 



"[(-) OOB {( + ) + zsin 0}- t cos {(n + {) 0-zsin 0}]d0 



o 



- sn n + iTr e--" nh * +( " +1/ ' ) *d-i sin 



* ' (Euvres Scientifiques,' tome L, p. 405. 



t Math. Ann. Band III, p. 143. 



J In what follows n will be taken to be a positive integer, and u and v will be written for u n and v n . 



