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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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    What is $g(z,0)$? From what you first wrote, it seems as if $g$ is meant to be a function of two *real* variables.2012-01-30
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    @AndresCaicedo : I think they changed the arguments of $g$. I do not fully understand that myself, hence I am asking... :) Maybe the new arguments are $(z,\bar{z})$? this is a wild guess though.2012-01-30
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    @AndresCaicedo : On second thoughts my suggestion of change of variable doesnt make sense since if $\bar{z}=0$ then so is $z$. Unfortunately this is the question as presented and I am rather confused...2012-01-30
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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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    Thanks, Robert! May I ask what do you mean by "has a limit point in $D$"? Also, would you mind perhaps giving me a link to a proof of this theorem?2012-01-30
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    Point $p$ is a limit point (or accumulation point) of set $S$ if for every $\varepsilon > 0$ there are infinitely many points of $S$ within distance $\varepsilon$ of $p$.2012-01-30
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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