OF ARBITRARY CONSTANTS, &c. 107 



The integral and the series are both convergent for all values of a, and either of them 

 completely defines u for all values real or imaginary of a. We easily find from either the 

 integral or the series 



du 



-r- + 2aM = 2 (2) 



da 



This equation gives, if we observe that m = <then a = 0, 



« = 2e-«^ e«da=2e-«Ja + — +^-^+^-^^ + ...} (3) 



This integral or series like the former gives a determinate and unique value to u for 

 any assigned value of a real or imaginary. Both series, however, though ultimately conver- 

 gent, begin by diverging rapidly when the modulus of a is large. For the sake of brevity I 

 shall hereafter speak of an imaginary quantity simply as large or small when it is meant that 

 its modulus is large or small. 



2. In order to obtain m in a form convenient for calculation when a is large, let us seek 

 to express u by means of a descending series. We see from (2) that when the real part of a* 

 is positive, the most important terms of the equation are Saw and 2, and the leading term of 

 the development is a~'. Assuming a series with arbitrary indices and coefiicients, and deter- 

 mining them so as to satisfy the equation, we readily find 



1 ] 1.3 



M = - + — - + 



a 2a' To/ 



This series can be only a particular integral of (2), since it wants an arbitrary constant. 

 To complete the integral we must add the complete integral of 



du 



-— + 2aM = 0, 



da 



whence we get for the complete integral of (2) 



1 1 1.3 1.3.5 



- + — + 1- 



a 2a' 2V a^a" 



M= Ce-"" + - + — - + -f- + -V^ + (4) 



This expression might have been got at once from (3) by integration by parts. It remains 

 to determine the arbitrary constant C. 



3. The expression (l) or (3) shews that « is an odd function of a, changing sign with a. 

 But according to (4) u is expressed as the sum of two functions, the first even, the second odd, 

 unless C = 0, in which case the even function disappears. But since, as we shall presently see, 

 the value of C is not zero, it must change sign with a. Let 



a = p (cos Q + v/— 1 sin 6). 



Since in the application of the series (4) it is supposed that p is large, we must suppose a 

 to change sign by a variation of 9, which must be increased or diminished (suppose increased) 

 by IT. Hence, if we knew what C was for a range tt of 0, suppose from Q = a to = o + 7r, 

 we should know at once what it was from = a -^ v to & = o+ 27r, which would be sufficient 



14—2 



