SPECTRA OF QUANTIZED SIGNALS 465 



By calculating the Jacobian of the transformation, we find dx dy = 2 du dv. 

 The region of integration in the uv-p\sine is a rhombus bounded by the lines 

 w zfc t) = rbl. We then have: 



v') 



^(") = 16.(1 -.Y/^ LI. L. -^^ + L '" L H ^"' - 



r y^/ «^ , / 1 v' -2mk{2u+2m) 



^^PL-iVrT-a + l^^g-o^P 4(1 +.) 



-^ -2»*(2s + 2«, ^ -(2ot+1)A(2m + 2»j + 1) 

 „?„^^P 4(1 - a) + J.^'^P 40+^^ 



t -(2n + iy(>.. + 2« + l) (,13) 



n=-oo 4(1 — a) 



If we substitute u = — ic in the first double integral, m = —m' in the first 

 series, and m = —m! — 1 in the third series, we see that the two double 

 integrals are equal. We therefore drop the first double integral and multiply 

 the second by two. The inner integral may then be split into parts with 

 limits from v = ^tov=\ — u and v = u— \tov = ^. Substituting v =—y 

 in the second part and treating the series as before, we find that the two parts 

 give equal contributions, so that the bracketed integral terms become 



/.I f'\—U 



\ du \ dv 



Jo Jo 



apphed to the integrand. 



The series in (2.18) may be written as Theta Functions, and the imaginary 

 transformation of Jacobi then used as an aid in reduction. We may 

 proceed in a more direct manner, however, by applying Poisson's Summation 

 Formula :^^ 



Z vi2irn) = ^ t, r <fiir)e-'-' dr (2.19) 



n=— 00 ^TT w»=— 00 •'—00 



We thereby show that 



JO 



2 exp [—am{x + 2m)] = 



Wl= — 00 



, /tT ax^lS fl 1 O V^ -m2/x22a ^^^ fKirxl . . 



\ TT ^ 1 + 2 2^ e cos — - (2.20) 



\ la L w.=i I J 



E exp [-fl(2w + l)(a: + 2m + 1)] 



m=— 00 



^ 1 A ^a..,4 [i ^. 2 f; (_ )•»« -"•''^'^- COS "^'\ (2.21) 



2 \ a L »»=i 2 J 



