Analytic Properties of Fourier Integrals

Let us first restrict ourselves to causal functions, which are mainly of interest. We have 

Let the integral exist for real co. This will be true for example if f(t) falls off as Of 1. Then, for all co in the lower half-plane, the integral is certainly convergent and/(co) is analytic. This is an immediate consequence of i he fact that t is non-negative in (A3.2.1). Furthermore, along any radius from the origin to infinity in the lower half-plane, f(co) will decay exponentially to zero. 

Consider now the converse of this statement. Let f(co) be analytic in the lower half-plane. Then we claim that/(0, defined by (A3.1.2), is zero fo 0. This is easily shown if we observe that for 0, we can close the path of integration in (A3.1.2) around the lower infinite half circle, which does not contribute because of the exponential decay factor in the integral. But this contour integral is zero. 

This result also applies when a 0, so that, if f(t) decays exponentially,/(co) is analytic in part of the upper half-plane as well as the lower half-plane. Thus, its region of analyticity is expanded. 

Consider now the converse of this statement. If f(oj) is analytic for Imcu - a, but not above this line, what can we say about /(/) If we define/(r) according to 

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