Notes on Complex Analysis

Complex analysis is an area of mathematics that is concerned with complex numbers, functions, and calculus. Complex analysis is a simple extension of the real-number calculus to the complex domain. We’ll apply the calculus concepts of continuity, derivatives, and integrals to the case of complex functions of a complex variable. We’ll come across analytic functions along the way, which are the focus of this introduction. In reality, complex analysis is mostly the study of analytic functions.

Any complex number z is a point in the plane (x,y), hence z = x+iy, where  i=-1. Any complex function of a complex variable z can be divided into two functions in the same way, as in f(z)=u(z)+iv(z), or, f(x,y)=u(x,y)+iv(x,y)  Because such functions include two separable functions and are reliant on two independent variables, visualising the function would require a four-dimensional space, which is difficult to conceive.

We will look at Complex analysis , Complex numbers, Complex Functions, and Complex Number Roots in this article.

Complex numbers

A complex number is one that has both real and imaginary components and may be represented as a + bi, 

where a and b are real numbers and i is the imaginary element, and the equation i2= 1 must be satisfied. The real and imaginary parts of a complex number are referred to as a and b, respectively.

Complex numbers enable solutions to all polynomial equations, including those with no real-number solutions. The fundamental theorem of algebra states that every non-constant polynomial equation with real or complex coefficients has a complex number as a solution. 

Because the square of a real integer cannot be negative, the equation (x+1)2=-9 has no real solution, but it does have two nonreal complex solutions, -1 + 3i and -1 -3i.

Complex Functions

A complex function is one that goes from one complex number to another. In other terms, it is a function with a domain of complex numbers and a codomain of complex numbers. A nonempty open subset of the complex plane is supposed to be present in the domain of complex functions.

The values z from the domain and their images f(z) in the range can be divided into real and imaginary parts for any complex function:

z=x+iy  and fz=fx+iy=ux,y+iv(x,y)

where x,y,ux,y,v(x,y) are all real numbers.

To put it another way, a complex function f:C→C can be broken down into

v:R2→R

Specifically, into two real-valued functions (u,v) of two real variables (x,y).

Some characteristics of complex-valued functions are identical to those of vector-valued functions with two real variables. Other complex analysis notions, such as differentiability, are direct generalisations of analogous real-function ideas, but they may have significantly different features.

Root of Complex Numbers

Another complex number whose square equals the provided complex number is the root of a complex number.

For example, if a + ib is (a + ib) = x + iy,  is the square root of a + ib, 

then  (x + iy)2 = a + ib is the result.

Comparing the real and imaginary parts of the equation (a + ib) = x + iy  squaring both sides and then calculating the values of x and y is one of the easiest techniques to get the square root of a complex number a + ib 

Conclusion

In this article we learned that, Complex analysis is a branch of mathematics that studies functions of complex numbers. It is sometimes known as the theory of functions of a complex variable. 

Many fields of mathematics, such as algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as physics, such as hydrodynamics, thermodynamics, and especially quantum mechanics, benefit from it. Complex analysis has applications in engineering domains such as nuclear, aeronautical, mechanical, and electrical engineering by extension.