ttiSMNDIKJMHOM 



there. (Here we assume that K(s) has no zeros in the strip D, a point 

 we shall examine again later.) It is remarkable that such a factoriza- 

 tion exists for any K(s) which is analytic and non-zero in D but which 

 may have any kind of singular behavior outside D; the proof was given 

 by Wiener and Hopf [1], Hera the truth of the theorem is obvious; 



V 8 > - (ttt) • K » ■ (rny < 3 - 2) 



is one factorization with the required properties, and is such that 



K (ft) ■* 1 at infinity in R . After we have completed the solution 



we thall return to the uniqueness or otherwise of this factorization. 



Because K (s) is free from zeros in R , and in particular in D, 



we can divide (2.26) through by it, to get 



Y (s) 

 K + (s) Y + (s) + ~— - ___ (3.3) 



- o - 



The analyticity properties of the terms on the left here are known, 



wh^le the function on the right is neither a@ function nor a (^function. 



Our next object is to write it as the 3JM of the two functions analytic 



in R and R respectively, and of algebraic behavior at °° in those 



half-planes. Again, the existence of this ADDITIVE SPLIT is assured 



by the W-H theorem [1], and here wc can again- see how to perform the split 



by inspection (and in this case the method of inspection is widely 



useful and should be carefully noted) . 



The function i/(s + k ) K (s) is analytic in R except for the 

 o — — 



pole at s ■ -k . Near the pole, the function behaves like 



i/(s + k ) K (-k ), so that we can isolate the pole behavior by adding 

 o — o 



and subtracting this term, to giva 



14 



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