Can anyone give a good example of a function that is real-analytic at some point but not complex-analytic there? I've already given one that is infintely differentiable but not analytic.
In [[complex analysis]], a branch of [[mathematics]], the term '''''analytic function''''' is often used synonymously with [[holomorphic function]], which see.
However, the term admits another meaning in mathematics. An '''analytic function''' is a function that is locally given by a convergent [[power series]].
If a function ''f'' is differentiable in some open disk ''D'' centered at a point ''c'' in the complex field, then it necessarily has derivatives of all orders in that same open neighborhood, and the power series
converges to ''f''(''z'') at every point within ''D''. This is an important respect in which complex functions are better-behaved than real functions. Consider the real function
One can show that ''f'' has derivatives of all orders at every point including 0. To show this at ''x'' = 0, use [[L'Hopital's rule]], [[mathematical induction]], and some simple substitutions. But in proving this, one will find that ''f''<sup>(n)</sup>(0) = 0 for all ''n''. Therefore, the [[Taylor series]] of ''f'' is zero at every point! Consequently ''f'' is '''''not analytic''''' at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable.
One may say of a real function of a real variable that it is ''analytic'' if it is locally equal to its Taylor series, regardless of whether it remains so if one attempt to extend it to a function of a complex variable. In that case, ''analytic'' ceases to be synonymous with holomorphic.
