Complex Analysis refers to the study of complex numbers. In dealing with Complex Analysis, it is important to understand the different terms. A complex number, in Mathematics, is a number that can be expressed in the form a+bi, where ‘a’ and ‘b’ are real numbers, and ‘i’ is the imaginary unit. It is also important to note that i^2=-1. Thus, a+bi is a complex number because it is a mix of real and imaginary, with the ‘a’ being real, and the ‘b’ being imaginary. Complex numbers are often depicted on a plot, where the x-axis represents the real value of the complex number (a) and the y-axis represents the imaginary value of the complex number (b). The point, which is plotted, is thus known as point (a, b). One thing to consider is that numbers written in the a+bi form are not necessarily complex numbers. For example, if a = 0 then the number is purely imaginary. If b = 0, then the number is completely real.
The creation of these complex numbers gives an additional tool to solve problems that cannot be solved with just real numbers alone. However, one important realization to note is that these imaginary numbers are no more or less fictitious than any other kind of number. With this in mind, these complex numbers have practical applications in many fields other than Mathematics – most prominently in electrical engineering. For example the imaginary ‘I’ is sometimes used to designate current in power systems. AC circuits in particular have their resistance and reactance denoted by two complex numbers. The sum of these two complex numbers is then known as the impedance, denoted by the symbol Z.
Thus, understanding complex numbers, and by extension Complex Analysis is a crucial skill to possess when dealing with complex problems.