Which theory specifically involves the study of complex numbers?

Complex Analysis is the branch of mathematics that focuses specifically on the study of complex numbers and their functions. It delves into the properties of complex numbers, which are numbers that consist of a real part and an imaginary part, and explores high-level concepts such as holomorphic functions, contour integrals, and residues. This field has a broad range of applications, including in engineering, physics, and applied mathematics.

While trigonometry deals with relationships between angles and side lengths in triangles, and linear algebra focuses on vector spaces and linear mappings, these subjects do not primarily focus on complex numbers. Vector theory involves concepts related to vectors in multi-dimensional spaces, and while it may touch upon complex numbers, it is not dedicated to their study. Therefore, Complex Analysis is the distinct area that thoroughly examines complex numbers and their applications.