📂Complex Anaylsis

Definition of a Complex Function

Definition ¹

For a non-empty subset $A,B \subset \mathbb{C}$ of the set of complex numbers $\mathbb{C}$, $f : A \to B$ is called a Complex Valued Function. On the other hand, when $A, B \subset \mathbb{R}$, $f : A \to B$ is also referred to as a Real Valued Function to distinguish it from complex functions.

Explanation

The above definition actually means nothing. You might wonder what all this is about, but if you start to nitpick, there are too many details in this definition to quibble over, making it worthless as a definition.

• The expression Real, Complex Valued Function technically distinguishes whether the ‘value of the function’ is real or complex, so it seems reasonable to define it regardless of the domain.
• Just because the codomain is $\mathbb{C}^{n}$, it’s not called a complex vector function. Naturally, the corresponding expression for $f : \mathbb{C} \to \mathbb{C}$ is not called a complex scalar function either.
• It does not call a function both a complex function and a real function just because its domain is real numbers and its codomain is complex numbers, or vice versa.
• In actuality, although real functions seem to refer to most of the functions encountered from middle and high school, in mathematics departments, many courses and textbooks also call topics related to measure theory simply Real Analysis, which can cause confusion.

The point is, the expressions complex function and real function are not used based on strict definitions but follow conventions depending on the context. For example, $f : \mathbb{C} \to \mathbb{C}$ is just called a complex function by everyone, and $f : \mathbb{R} \to \mathbb{C}$ can be called a complex function, but there is a ’tendency’ to distinguish it as ‘a function whose value is complex’. $f : \mathbb{C}^{n} \to \mathbb{C}^{n}$ is universally called a complex function, but it is rare to call $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ a real function, mainly because $\mathbb{C}^{n}$ is meaningful as a generalization of some function or theorem, but $\mathbb{R}^{n}$ tends to have a strong implication as a vector function.