B. Representation theorems for functions analytic in a strip
Let 01 be a curve in the domain DO (defined in section 5) of the complex &plane defined by Re VP + b = cl, where cl > 0 and cl is such that 01 does not intersect the cuts of Do. Then Re il is bounded on 01. If b is not on the negative real axis or zero and 0 0 and 02 does not intersect the cuts of Da. Now we suppose that cl and cs are chosen in such a manner that 01 and 0s are in DO and do not intersect each other and the cuts. Let G be the open subset of DO where c2 > Re VP + a, Re VP + b < cl or cs < Re 11s + a, Re m<ci.
Then the boundary of G consists of 01 and 0s and G consists of one or two subdomains. The real part of 2 is bounded on G. There are three types of sets G.
Type I. ci2Re l/b and csh Re Ia. Then G consists of one or two strips extending from Im 1= -00 to Im fl= + co. Two strips occur in the case that none of the numbers a and b is negative or zero. Then there is a strip to the right and one to the left of the cuts, with boundaries of type As or A6 (cf. fig. ). In the case that one of the numbers a and b is negative or zero there is only a single strip to the right of the cuts.
Type II.
Neither of the numbers a and b is on the negative real axis or zero and 0 <cl 5 Re I/b, 0 <ca dRe ia. (In this case cl = Re l/b and cs = Re l/a cannot occur simultaneously since then 01 and 0s would intersect at the origin). There are now two strips, one in the upper half plane