I want to understand the relation between Holomorphism and Smoothness.
I want to elaborate the question as there are some underlying intricacies involved in the definitions:
Smooth: A function is smooth if it is infinitely differentiable at every point of its domain. ( I have mostly heard this definition when speaking of Real functions)
Holomorphism: A function is holomorphic if the function is differentiable at every point in the neighbourhood. (I have never heard this term when reading Real analysis)
Analytic: A function is analytic if its power series representation equals the value of the function at that point.
I know the subtle difference 1) Smoothness does not imply Analyticity for Real Analysis 2) Holomorphism implies Analyticity for Complex Analysis
I want to know if Holomorphism and Smoothness are one and the same thing. Is it that they are just two different notions where one is used in Complex Analysis and the other in Real Analysis.