Hi Again: I don't know if this is obvious or not: An analytic complex function f(z)=u(x,y)+iv(x,y) , can be made into an analytic function f: R^2 -->R^2, since each of u(x,y) and v(x,y) is itself a real-analytic function, i.e., we can use a standard argument by component function. How about in the opposite direction, i.e., we have a real-analytic function f(x), analytic in an interval (a,b). When can we extend f(x) into a complex-analytic function.?. I suspect , thinking of power series, that we can use the radius of convergence to construct an analytic function, i.e., if f(x) is analytic in (a-r,a+r), then f(x) can be extended to a complex-analytic function in |z-a|<r . Is this correct.?