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I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique.

Now i am wondering if this is true for real functions. I mean, if $f: \mathbb{R} \to \mathbb{R}$, when is it true that there is an analytic $g$ whose restriction to $\mathbb{R}$ coincides with $f$ and also when is $g$ unique.

Surely $f$ needs to be differentiable but this might not be sufficient for existance of such $g$.

edit: I mean, is it easy to see that there is and extension of sine cosine and exponential real functions?

Thanks a lot.

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  • $\begingroup$ Wait--what? That's not true, is it? Are you talking about analytic functions $\mathbb C\rightarrow\mathbb C$? What about $z\mapsto 1/z$ on a small disc around $z=1$? Or $\log z$ in the same disc? $\endgroup$
    – MPW
    Jun 20, 2014 at 1:04
  • $\begingroup$ This is not nearly a strong enough condition. You need that $f$ must be real analytic to even hope to analytically continue it. $\endgroup$
    – Cameron Williams
    Jun 20, 2014 at 1:05
  • $\begingroup$ Your first sentence is terribly false. And, yes, you need $f$ to be real analytic to have a hope. But what do you do with $f(x)=\dfrac1{1+x^2}$? $\endgroup$
    – Ted Shifrin
    Jun 20, 2014 at 1:06
  • $\begingroup$ @TedShifrin: Yes, agreed! That's what I meant in my comment. That can't possibly be correct. Analyticity is a rather rigid condition. $\endgroup$
    – MPW
    Jun 20, 2014 at 1:08
  • $\begingroup$ Oooops sorry, I am editing it. What a terrible sentence $\endgroup$
    – ThePortakal
    Jun 20, 2014 at 1:08

2 Answers 2

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Necessary and sufficient condition for existence of an entire function $g$ extending $f$ to the whole complex plane: $f$ is infinitely differentiable at $0$, and the power series for $f$ at the origin converges to $f$ on the real line.

A counterexample for something more is (as noted in a comment) $f(x) = 1/(1+x^2)$. It is real analytic on the real line, but cannot be extended analytically to any connected region containing both $0$ and either $i$ or $-i$. The power series at $0$ only has radius of convergence $1$.

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  • $\begingroup$ Is there an example of a function on $\mathbb{R}$ having a nonunique extension to $\mathbb{C}$? $\endgroup$
    – John Fernley
    Apr 29, 2016 at 14:29
  • $\begingroup$ say if a holomorphic function is identically $0$ on $\mathbb{R}$ is it $0$ on the whole plane? The dirichlet problem implies this is true but I wasn't confident as $\mathbb{R}$ is closed $\endgroup$
    – John Fernley
    Apr 29, 2016 at 14:32
  • $\begingroup$ @JohnFernley ... That is the "identity theorem" mentioned before. If two analytic funtions defined on a connected domain agree on a set with a limit point (for example, the real line is such a set), then they agree on the whole domain. $\endgroup$
    – GEdgar
    Apr 29, 2016 at 14:32
  • $\begingroup$ that's quite straightforward, thanks. I had forgotten that condition $\endgroup$
    – John Fernley
    Apr 29, 2016 at 14:33
  • $\begingroup$ Is the class of real-valued functions extensible to entire functions a good class to "approximate" the class of analytic functions? And for infinite differentiable functions? $\endgroup$
    – Canjioh
    Dec 9, 2020 at 21:23
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I think that most mathematicians would say that the functions you mention are restrictions to the real line of functions more naturally defined on the complex plane in the first place. So--yes.

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