where q is the greateat integer < — 



= 2 



Q t 



— anti r is the greatest integer < 1 — 



2x =22 



183 

 Q 



(tnd conversely, every periodic function of period 1 which can be written in the form 



(2) i'.s analytic in the finite plane and satisfies a relation of the form (1). 



Since p(x) is periodic of period 1, it takes in any period strij) all the 



values it takes anywhere in the finite plane. The transformation 



2xix 

 \v = e 



carries a single period strip of the x-plane into the whole w-plane, z' = + ^ 



corresponding to w = 0. and z' = -oc to w = x . 



We can now write 



pfx) = f(w) 



and since f(w) can have only the singular points zero and infinity it is ex- 

 pansible in a Laurent series 



f(w) = S Biw^ 



valid throughout the finite plane except at zero. 



Using the fact that 



— 2Ttz' 



we get 



I 

 |p(x) e 



when z' is positive, and 



I t Q 



-txlz']— QzM 1 1 



I = I f (w) w 2 2x 



2-r: 



|p(x) e I = |f(w) w 



when z' is negative. As z' = + ^ . w = and 



I t Q I 



L - + — I 



w = o |f(w) w 2 2ti: I = b. 



Hence the part of the series f(w) with negative exponents can not have co- 



t Q 



efficients different from zero for j greater than the greatest integer < 1 . 



= 2 2 X 

 As z' = — X, w = 00 and 



(t Ql 



L 



w = X f(w) w [2 2xJ = b. 



Hence the part of the series f(w) with positive exponents can not have co- 



t Q 



efficients different from zero for j greater than the greatest integer < . 



= 2 2x 



