SECT. IV.] REAL AND UNREAL PARTS OF A FUNCTION. 43.&quot;) 



same function ; for the summation of the series is reduced, by 

 what precedes, to that of a series of algebraic terms. 



420. We may also employ the theorem in question for the 

 purpose of substituting under the general form of the function a 

 binomial formed of a real part and an imaginary part. This 

 analytical problem occurs at the beginning of the calculus of 

 partial differential equations ; and we point it out here since it 

 has a direct relation to our chief object. 



If in the function f(x) we write \L + v 1 instead of #, the 

 result consists of two parts (b+Jlty. The problem is to 

 determine each of these functions &amp;lt;/&amp;gt; and ty in terms of //. and v. 

 We shall readily arrive at the result if we replace f(x) by the 

 expression 



for the problem is then reduced to the substitution of /A + v I 

 instead of x under the symbol cosine, and to the calculation of the 

 real term and the coefficient of 1. We thus have 



=/(/* + v J~l) = ~jdz (*) fdp cos [p (p - a) +pv 

 4~ p a /( a ) I 



cos ~* e pv + e ~ pv 



l sn - 



hence $ = |d/(a) [dp cos (pp -pz) 



Thus all the functions f(x) which can be imagined, even those 

 which are not subject to any law of continuity, are reduced to the 

 form M -f- Nj 1, when we replace the variable x in them by the 

 binomial yu,+ v*J- 1. 



282 



