182 



We make use of the following form of Stirling's fornnila: 



X — Xl — 5 — X + Xi 



|(X — Xl) = (x — X,) c v'2x (1 + Ex), 



where e.\ approaches zero as x approaches infinity in such way that its distance 



from the negative axis of reals approaches infinity. 



Then we have 



I _ — X + Xl + ^ X — Xl I — 



lim |i(x — Xl) . (x — Xl) e ! = c. 



X = 00 I j 



Set X = z + iz' and xi = Ui + ivi, where z, z'. Ui, Vi are real, and let x approach 



infinit}', A < R (x) < A + 1; then we have 



lim 



— z — iz' + U| + ivi + 2 z + iz' — Ui — iv 

 (x — Xl) . (z + iz' — Ui~ivi) e 



Hence 



c, 



lim I. (— z— iz' + u, + ivi + i) (log ,(z— u,)=+(z'— vi)=+iOi) 



zi==fcx ||(x — Xl) . e 



Z — Ui' 



e 

 where 



z" — v, 

 Oi = tan-' . 



Z — Ui 



Now z — Ui >(), Iherefore when z' = + ^c , Oi = — and when z' = oo , 



2 



Oi = — . Thus in the above limit after mulitplving the factors in the 



2 



exponent of e, we can replace z'O, by — |z'| . Then by reai-rangcmcnt and sim- 



plification we can write 



lun 

 z 



Z + Ui + ^ Iz'l Z Ui ^ Oi Vi 



e 2 e 



=>= X I |(x Xl). z' 



Making use of limits of this form for each of the ganuna functions in the 

 expression in the lemma, we have the lemma. 



Lemm \ II, // p (x) is (I periodic funclion of {>crio<l 1 ii'hirh is nnalyiic 

 everywhere in the finite plane and as z' = =t x (x = z + iz') satisfies the relation* 



(1) L I — tx|z'l — Qz'l 



z' = ± cc |p(x) e I = b, 



h finite, t positive, then p(x) may be written in the form 



c] 2xijx 



(2) p(x) = i: Bje 

 j = -i' 



*I, 1 -^ ± x <lonotes the greiteU value upproachcil as z' r==: ± ^- 



