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I have been asked the following question and would appreciate an explanation.

Suppose we have to find an analytic function $F(z)$ where $z=x+iy\in \mathbb C$ and its real part is $g(x,y)$. Question: Does it suffice to be given $F(z)=g(z,0)$ for us to determine $F(z)$ in general?

I am not sure though I guess the fact that analytic functions only depend on $z$ might be relevant? (I might be wrong though!)

Thank you.

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    Maybe the problem is to find an analytic function $F(z)$ with real part $g(z)=g(x,y)$, with an other condition like $F(0)=g(0)$. However, you should rewrite it if you want an answer.2012-01-30

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I think they mean $F(x) = g(x,0)$, i.e. on the real axis, the analytic function is real, and its values are given there. The answer is yes, by the uniqueness theorem for analytic functions: any two analytic functions on a domain $D$ that are equal on a set that has a limit point in $D$ are equal everywhere in $D$.

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    ... but this is a level of generality that you probably don't need at this point. More simply: if $F_1$ and $F_2$ are two possible choices for $F$, then since the derivatives at a point on the real axis can be computed using only the values on that axis, we find that $F_1$ and $F_2$ have the same Taylor series about any point on the real axis.2012-01-31