Jtdylktuy

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.

At a first glance, it sounds like smoothness (having all derivatives) implies holomorphy (having one derivative). However, there are two different meanings of derivative being used here. A complex function $f:mathbb Cto mathbb C$ is smooth if, when considered as a function $f:mathbb R^2to mathbb R^2$, both component functions of the output have all higher order partial derivatives. On the other hand, a complex function $f$ is called holomorphic if it is complex differentiable, meaning its first partial derivatives exist and satisfy the Cauchy-Riemman equations.

To see these are not equivalent, note that $zmapsto overline{z}$ is smooth (it is a linear map $mathbb R^2to mathbb R^2$), but not holomorphic.

It is true that every holomorphic function is smooth.

We can parametrize $mathbb{C}$ with a complex coordinate $z$ or a pair of real coordinates $x+iy$ where $z=x+iy$. This lets us look at points in $mathbb{C}$ as pairs of real numbers $(x,y)$ without any serious issues. It turns out that "complex differentiability" of $f$ at $z$ i.e. the condition that
$$ f'(z)=lim_{hto 0}frac{f(z+h)-f(z)}{h}$$
exists is very strong. Using this we can prove that $f$ viewed as a function $f(x,y)=u(x,y)+iv(x,y)$ from $mathbb{R}^2to mathbb{R}$ satisfies the Cauchy-Riemann equations:
$$ begin{cases}
partial_xu=partial_yv\
partial_y u =-partial_x v.
end{cases}$$
Developing the theory of functions of a complex variable, we can prove that any complex differentiable (holomorphic) function $f$ on an open domain $Omegasubseteq mathbb{C}$ is analytic. That is, it can be written as a convergent power series locally around any $z_0in Omega$ as
$$ f(z)=sum_{k=0}^infty a_k (z-z_0^k).$$
Such functions are called analytic. On the other hand, there exist smooth non-analytic functions in the real case. For instance take the bump function $g:mathbb{R}^2to mathbb{R}$ given by
$$ g(x,y)=
begin{cases}
expleft(frac{-1}{1-(x^2+y^2)}right)& x^2+y^2<1\
0 & text{else}.
end{cases}$$
This function is infinitely many times differentiable (as you can check), but does not have a convergent power series expansion. The fact that such bump functions exists makes the real case of manifold theory a bit more flexible, while the complex situation is a bit more rigid. The difference is that complex differentiable functions also have to satisfy a system of partial differential equations, in essence.