Complex analysis is a branch of mathematics that deals with functions of complex numbers. It is important because it has many applications in physics, engineering, and other areas of science, and it provides a powerful framework for understanding and solving problems in these fields.
A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex numbers are often used to represent quantities that have both a real and an imaginary component, such as electrical currents and waves.
A real function is a function that takes real numbers as inputs and outputs real numbers. A complex function is a function that takes complex numbers as inputs and outputs complex numbers. In other words, a complex function is a function of a complex variable, while a real function is a function of a real variable.
The Cauchy-Riemann equations are a set of two partial differential equations that describe the conditions for a function to be differentiable at a point in the complex plane. They are important in complex analysis because they provide a way to determine if a function is analytic (or holomorphic), which means it can be represented by a power series and has many useful properties.
One example of a complex analysis problem is finding the maximum value of the function f(z) = z^2 + iz on the unit disk (the set of all complex numbers with absolute value less than or equal to 1). The solution involves using the Cauchy-Riemann equations to show that f(z) is analytic on the disk, and then using the maximum modulus theorem to find the maximum value on the boundary of the disk. The solution is f(z) = 1 + i, which occurs at the point z = 1/√2 + i/√2.