Terminology: Analytic, Angle-preserving, Conformal and Holomorphic

Question

I am trying to contrast the relations between Terminology:

1. Analytic,

2. Angle-preserving,

3. Conformal

4. Holomorphic

My question for you is that how "Analytic, Angle-preserving, Conformal and Holomorphic" are related to each other? Say in complex analysis or other discplines?

For example, in this video "Visualizing the Riemann hypothesis and analytic continuation ": https://www.youtube.com/watch?v=sD0NjbwqlYw , it says at 13:57 that

complex function with derivative everywhere is called analytic. Analytic is almost about Angle-preserving

Analytic $\Leftrightarrow$ Angle-preserving (kind of)

However, analytic of complex function is not entirely about angle-preserving (kind of).

Answer 1

Conformal and angle-preserving are the same thing, and refer to maps that preserve the angles between intersecting curves and the shapes of infinitesimally small objects. This can be applied to any mapping between manifolds, and is not restricted to $\mathbb C$ or $\mathbb R$.

Analytic functions can be represented in terms of a locally convergent power series, and are thus infinitely differentiable. Typically these are considered as functions of a real or complex variable, though there's probably generalizations to other measure spaces. Like with continuity and differentiability, a function can be analytic at some points and not at others.

Holomorphic functions are complex differentiable. In this case we're really only talking about functions $\mathbb C \rightarrow \mathbb C$, or some subsets thereof. Functions can be holomorphic at some points and not others. A function that is holomorphic on all but a set of isolated points is called meromorphic, and a function that is holomorphic on all of $\mathbb C$ is called entire.

How are they related? Well, a major theorem of complex analysis is that if a function is holomorphic, then it is analytic. This is rather different from functions of a real variable, where analyticity is a far stronger condition than simple differentiability. Further, a holomorphic function is a conformal map anywhere its derivative is nonzero (this extra condition about being nonzero might be what the video is referring to when it says "kind of"). So in complex analysis, it turns out all these properties describe the same kind of function.