given any function (or maybe distribution) f(x) and g(x) so [tex] f(x+i\epsilon ) - f(x-i\epsilon ) = g(x) [/tex] if we know f(x) could we obtain g(x) from the difference above ?? if we knew g(x) could we solve the equation to get f(x) ?? here [tex] \epsilon \rightarrow 0 [/tex]
For the first question, if you know f(z) for all complex z, you can get g(x). For the second question the answer is no, since you can add any continuous function to f(z) and still get the same g(x). I presume that f(z) is discontinuous perpendicular to the real axis, otherwise the question is trivial.