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Let z=x+iy.

I have an equation of the form f(x)=g(x).

Under what conditions, if any, is it true (if it IS true) that f(z)=g(z) for all z?

It looks like analytic continuation, but not quite, since analytic continuation begins with the supposition that f(z)=g(z) over the region y=0, and I'm not sure that that is the same thing.

I vaguely recall a theorem from my long-ago undergraduate days that says something about when it is allowed to replace x by z in the argument of a function, but cannot find it now. Can someone give me a reference? Thank you.

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    It is the same thing: when $y=0$, $z=x$. – 2017-01-24
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    If $f$ and $g$ are both analytic, then $f(x)=g(x)$ for all $x\in \Bbb R$ (or even much weaker assumptions) implies that $f(z)=g(z)$ for all $z\in\Bbb C$. (Idetity theorem) – 2017-01-24

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