복소 해석학, or complex analysis, is a branch of mathematics that studies functions of complex numbers. It explores properties and behaviors of these functions, which can be more intricate than those of real numbers. Key concepts include holomorphic functions, which are complex functions that are differentiable, and Cauchy's integral theorem, which provides powerful tools for evaluating integrals in the complex plane. This field has applications in various areas, including engineering, physics, and applied mathematics. Complex analysis helps solve problems involving fluid dynamics, electrical engineering, and even quantum mechanics, making it a vital area of study in both theoretical and practical contexts.