Complex analysis is a branch of mathematics that describes functions which are complex in nature. For example, an electrical signal may be described as voltage or current and these signals normally consist of sinusoidal oscillations whose amplitudes and phases vary with time, so they are considered to be complex functions. This brings us to the concept of the complex plane. The complex plane is a two-dimensional grid of points, each with an x and a y coordinate.
On this grid, functions such as voltage or current can be plotted and analyzed to yield results that explain the behavior of real-world phenomena.
Complex analysis often uses the analytic techniques developed in the course of study (calculus) of real-world phenomena. This involves concepts such as continuity, differentiability, and integrality. Complex analysis is also often used to solve problems in physics and engineering, for example, fields like signal processing, optics or acoustics.
A complex variable can be thought of as a number which can be written in the form a + bi where:
a is some real number b is an imaginary unit i is the square root of -1
There are many ways to write complex numbers, but this one emphasizes that it can be viewed as a point in 2-d plane. From now on we shall denote complex numbers by z = x + yi.
A complex variable can be a function of real numbers, and the corresponding graph is called plane curve. A complex function is defined as an expression f(z), where z is in the domain of f. The equation for such a function may be written as y = f(x + iy). We know that every complex number has a conjugate pair. The conjugate pair for z = x + iy is (x – iy). It can be noted that the sum of the squares of the real parts of these numbers are equal to the square of their absolute values.
For instance, solving z^2 + 1 = 0 we find: z^2 = -1 and by squaring the real parts we get (x – iy)^2 = x^2 + 2xyi + y^2.
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The most common questions are:
What is the real number system?
How many dimensions does a complex plane have and what are they called?
When is a polynomial considered to be of 1st order, 2nd degree or some higher power?
What is a periodic function and how do you find it?
Who discovered complex numbers in mathematics textbooks?
It usually isn’t something you learn in school. A lot of students are stuck on an elementary level because teachers find it awkward to explain complex numbers and polynomials. Complex analysis is incredibly useful for mathematical modeling, especially quantum mechanics, which is impossible to do without it. There is no better way to solve these problems than asking someone who knows what they’re doing.
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