Am I correct in noticing that Complex Analysis seems to be a synonym for analysis of functions R2→R2?

If this is the case, surely all the results from complex analysis carry over to the study of these R2→R2 functions. Does anything from complex analysis carry over into the study of functions in even higher dimensions?

Furthermore, is there an area of Mathematics similar to complex analysis that investigates functions, say, R3→R3 to the same level of detail?

**Answer**

No, it isn’t. This can most obviously be seen from the fact that the map z↦¯z is not differentiable as a map from C to C, but the corresponding map (x,y)↦(x,−y) *is* differentiable as a map from R2 to R2.

This is because to be complex differentiable you need to have a best *complex* linear approximation which is a much stronger requirement than to have a best real linear approximation.

**Attribution***Source : Link , Question Author : providence , Answer Author : kahen*