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Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not any real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function ''f'' (''x'') defined in the paragraph above is a counterexample.
In particular, note that for a complex coordinate ''z'', the complex function that gives the real part of ''z'', namely Re ''z'' = (''z'' + ''z''<sup>*</sup>)/2, is '''not''' a complex analytic function. This follows, as a complex analytic function cannot depend on the [[complex conjugate]]. This is true as well for the imaginary part Im ''z'' = -i (''z'' - ''z''<sup>*</sup>)/2. In particular, one does not get a
It also follows that complex analytic functions are [[harmonic function|harmonic]], whereas real analytic functions in general are not. That is, the [[Laplacian]], when applied to a complex analytic function, vanishes. This is most easily seen by writing the Laplacian in complex coordinates:
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