jbunniii said:Brown and Churchill, "Complex Variables and Applications" is a good starting point. Consider also taking a look at Needham, "Visual Complex Analysis," which is not a traditional presentation of the material but rather a geometric way of looking at the subject, which can often be very insightful.
A complex variable function is a mathematical function that takes in complex numbers as inputs and outputs complex numbers. It is written as f(z), where z is a complex variable. This type of function is commonly used in mathematics, physics, and engineering.
The main difference between a complex variable function and a real variable function is that a complex variable function takes in complex numbers as inputs, while a real variable function takes in real numbers as inputs. Additionally, the outputs of a complex variable function are also complex numbers, while the outputs of a real variable function are real numbers.
Complex variable functions have many applications in mathematics, physics, and engineering. They are used to solve problems in complex analysis, differential equations, signal processing, and fluid dynamics, among others. They are also used in the study of fractals and chaos theory.
The Cauchy-Riemann equation is a set of two partial differential equations that must be satisfied by a complex variable function in order for it to be differentiable. These equations relate the real and imaginary parts of a complex variable function and are important in the study of complex analysis and functions.
A complex variable function cannot be graphed in the traditional sense, as its inputs and outputs are complex numbers. However, it is possible to visualize a complex variable function using color mapping, where different complex numbers are represented by different colors on a plane. Additionally, the domain and range of a complex variable function can be visualized using contour plots.