Complex analysis is a branch of mathematics that studies functions of complex numbers. A complex number consists of a real part and an imaginary part, typically expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. This field explores properties and behaviors of complex functions, including concepts like continuity, differentiability, and integration. One of the key results in complex analysis is Cauchy's integral theorem, which states that the integral of a holomorphic function over a closed curve is zero. This theorem leads to powerful applications, such as residue theory, which helps evaluate complex integrals and solve real-world problems in physics and engineering.