I know that in complex analysis there is holomorphic function and an anti-holomorphic function. The definitions of derivative for both functions are subtly different.

As a non-mathematician, I would like to know if there is a way to "analyze" complex functions which are neither holomorphic not anti-holomorphic? What are they known as?

For example the function z+(e^(z*)) is neither holomorphic nor anti-holomorphic, even though it has no singularities.


Knowing holomorphic and antiholomorphic functions does not exhaust all the theory of functions on $\mathbb C$. If you identify $\mathbb C$ with $\mathbb R^2$, holomorphic functions are functions that are differentiable in all directions and satisfying the Cauchy-Riemann equations. However, there are functions which are differentiable in all directions and not satisfying Cauchy-Riemann, functions which aren't differentiable, not continuous and so on. Holomorphic and antiholomorphic functions are just one important class that is well understood but there exist much more functions and they don't need to share any specific property.

  • $\begingroup$ Is there a name given to such functions, just curious. I had these ideas without reading a textbook, so I do not know where I can search for it. $\endgroup$ – Chetan Waghela Dec 6 '18 at 16:38
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    $\begingroup$ What I tried to explain is that there are many things with different names "in between". For example the function $z+e^{\overline z}$ you mentioned in your question is continuous, differentiable and things like that. You can define a class of functions of the form holomorphic + antiholomorphic. The question is whether this is then an interesting class of functions. But you for example also don't have a name for functions which are not smooth and likewise there is no special name for functions which are neither holomorphic nor antiholomorphic. $\endgroup$ – user526015 Dec 7 '18 at 8:15

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