182 



\\'{' make use of the following form of Stirling's formula: 



X — Xi — 2 X + Xi _ 



ifx — X,) = (x — X,) e v'2t: (1 + Ex), 



where Ex apjiroaches zero as x api)roaehes infinity in such waj' that its distance 



from the n(>gative axis of reals appi'oaches infinity. 



Then we have 



. |__ — x + x, + i X — x,| — 



lim Iji'x — Xi) . (x — xi) e ~ ''• 



X = 00 I 



Set x = z + iz' and Xi = Ui + ivi, where z, z', Ui, v, are real, and let x approaoli 

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



urn 

 z'= =fcoo 



Hence 

 lim 



— z — iz'+Ui + ivi+2 z+iz' — Ui — ivJ 

 (x — xi) . (z + iz' — Ui— ivi) e | 



■ =c I [ (x — x 1 ) . e 



(— z— iz> + u,+iv, + J) (log ,'(z— "i)=+(z'— v.)=+iO,) 



Z— Uil _ 



where 



Oi = tan- 



Now z — Ui >0, tlierefore when z' = + ^' , Oi = — and wlien z' = x, 



2 



Oi = — . Thus in the above limit aftei- muliti)lving tlie factors in the 



2 



X 



exponent of e, we can replace z'Oi by — |z'|. Tlien bv rearrangement and sim- 



2 

 plification we can write 



z + u 1 + 2 — I z ' I z — u 1 — 1 \' 1 I _ 



z ' = ± X I ! (x — Xi ) . z ' e :i e i ■ - 



Making use of limits of this form for eacli of the gannna functions in the 

 exjjression in the lemma, we ha\e the lemma. 



Lkm.ma II. // ]) (x) is It piyr iodic function of period 1 which is analytic 



everywhere in the finite plane and as z' = =*= oo (x = z + iz') satisjics (he relation* 



(1) L I — tic|z'| — Qz'l 



z' = ± X |p(x) e 1 = b, 



\) Unite, t /Kisiliee, then p(x) may lie written in tin form 



<1 2xiix 



(2j p(x) = i: Bje 



.i = -!• 



*L 1 -^ ^ X 'It'ioli's tlic Kreitcft value iipproaclicil us z' ;=z ± sc. 



