On the real plane (xy plane) inverse functions are reflections of their original functions over y=x. Is there such line for complex functions and their inverses?
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The linear mapping $(x,y)\mapsto(y,x)$ preserves handedness (as a real vector space), when it is mapping from $\Bbb{C}^2$ to itself, but reverses the handedness as a mapping from $\Bbb{R}^2$ to itself. A reflection w.r.t. any hyperplane of a real vector space OTOH always reverses handedness (its determinant $=-1$). – Jyrki Lahtonen Oct 28 '13 at 06:51
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Since we have more dimensions to deal with when working with functions of a complex variable, you cannot expect there to be a reflection in a line, but if you consider the 4-dimensional space needed to draw a "graph" of a function $w=f(z)$, then you should find that you get something similar by doing a reflection in the plane $w=z$.
The terminology can be a bit confusing: for a graph of $w=f(z)$, you need 2 real dimensions or one complex dimension for domain ($z$) and 2 real dimensions, or one complex dimension for the range ($w$). It is because of the 4-dimension nature of things here that we tend not to attempt to draw graphs of complex functions of a complex variable, but rather work with things like mappings between the $z$ plane and the $w$ plane, as they are easier to visualise.
Old John
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