In holomorphic analysis, complex numbers are considered one-dimensional because they can be represented as points on a one-dimensional complex number line. This is in contrast to real numbers, which can be represented on a linear real number line.
In the context of holomorphic functions, which are functions that are complex differentiable in a neighborhood of every point in their domain, the complex plane is the natural domain of study. A complex number can be written as z = x + iy, where x and y are real numbers and i is the imaginary unit (√(-1)). The complex plane is a two-dimensional space, but holomorphic analysis focuses on the behavior of functions defined on this plane.
When studying holomorphic functions, one-dimensional complex numbers are typically used as input or output values. For example, given a holomorphic function f(z), where z is a complex number, the function takes a complex number as an input and produces another complex number as an output. The complex number z is a point in the complex plane, which is effectively one-dimensional in the context of holomorphic analysis.
The reason for this focus on one dimension is related to the nature of complex differentiability. In one dimension, complex differentiability imposes strong constraints on the behavior of functions. In fact, the Cauchy-Riemann equations provide a necessary and sufficient condition for complex differentiability in terms of partial derivatives with respect to the real and imaginary parts of a complex variable. This allows for the development of powerful techniques and theorems in holomorphic analysis.
In summary, while the complex plane is a two-dimensional space, the study of holomorphic functions typically involves considering complex numbers as one-dimensional points on the complex number line. This is because the behavior of holomorphic functions can be analyzed effectively using one-dimensional complex numbers, allowing for the application of important results and techniques in holomorphic analysis.