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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.?

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# Real- and Complex- Analytic Functions.

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