Subjective Questions
Mathematical Analysis: Complex Analysis
Chapter 1: Introduction to Complex Analysis
In this chapter, we will delve into the fascinating world of complex analysis, a branch of mathematics that deals with complex numbers and their functions. Complex analysis plays a crucial role in various fields such as physics, engineering, and economics. This chapter will provide an in-depth introduction to complex analysis, covering the fundamental concepts and techniques used in this field.
Section 1: What are Complex Numbers?
In this section, we will start by defining complex numbers and exploring their properties. A complex number 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, defined as the square root of -1. We will discuss the real and imaginary parts of a complex number, as well as operations such as addition, subtraction, multiplication, and division of complex numbers.
Example 1: Simple
Simplify the expression (2 + 3i) + (4 – 5i).
Solution:
To simplify this expression, we add the real parts and the imaginary parts separately.
(2 + 4) + (3i – 5i) = 6 – 2i
Section 2: Complex Functions
In this section, we will introduce complex functions, which are functions that take a complex number as an input and produce a complex number as an output. We will explore the concept of a complex plane, where complex numbers can be represented as points in a two-dimensional space. We will also discuss the graphical representation of complex functions using contour plots and the concept of analyticity.
Example 2: Medium
Find the complex function f(z) = z^2, where z = x + yi.
Solution:
To find the complex function, we substitute z = x + yi into the equation and simplify.
f(z) = (x + yi)^2 = x^2 + 2xyi – y^2
Therefore, the complex function f(z) = x^2 + 2xyi – y^2.
Section 3: Complex Integration
In this section, we will explore the concept of complex integration, which is the counterpart of integration in real analysis. We will discuss the properties of complex integrals and the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. We will also introduce the concept of contour integration and the Cauchy\’s integral theorem and formula.
Example 3: Complex
Evaluate the complex integral ∫(2z + 3) dz, where the contour C is the circle |z – 2| = 1.
Solution:
To evaluate this complex integral, we parameterize the contour C and substitute it into the integral.
Let z = 2 + e^(it), where 0 ≤ t ≤ 2π.
dz = i e^(it) dt
∫(2z + 3) dz = ∫(4 + 2e^(it) + 3i e^(it)) i e^(it) dt
= ∫(4i + 2i e^(it) + 3 e^(it)) dt
= 4i t – 2 e^(it) + 3i e^(it) + C
Substituting the limits of integration, we get:
= 4i (2Ï€) – 2 e^(2Ï€i) + 3i e^(2Ï€i) – 4i (0) – 2 e^(0i) + 3i e^(0i)
= 8Ï€i – 2 + 3i – (-2) + 3i
= 8Ï€i + 2 + 3i
Section 4: Power Series and Laurent Series
In this section, we will discuss power series and Laurent series, which are representations of complex functions as infinite series. We will explore the convergence properties of power series and Laurent series and their applications in evaluating complex integrals and solving differential equations.
Example 4: Simple
Find the power series representation of the complex function f(z) = e^z.
Solution:
To find the power series representation, we expand the function into a Taylor series.
f(z) = e^z = ∑(n=0 to ∞) (z^n / n!)
Therefore, the power series representation of f(z) = e^z is ∑(n=0 to ∞) (z^n / n!).
Section 5: Residue Theory
In this section, we will introduce residue theory, which is a powerful technique used to evaluate complex integrals. We will discuss the concept of residues, which are the residues of a complex function at its singular points. We will also explore the residue theorem and its applications in evaluating real integrals, calculating infinite sums, and solving differential equations.
Example 5: Complex
Evaluate the complex integral ∫(1 / (z^2 + 1)) dz, where the contour C is the unit circle |z| = 1.
Solution:
To evaluate this complex integral using residue theory, we identify the singular points of the integrand and calculate their residues.
The singular points are z = i and z = -i.
Residue at z = i:
Res(i) = lim(z→i) ((z – i) / (z^2 + 1)) = (i – i) / (i^2 + 1) = 0
Residue at z = -i:
Res(-i) = lim(z→-i) ((z + i) / (z^2 + 1)) = (-i + i) / (-i^2 + 1) = 0
By the residue theorem, the integral is equal to 2Ï€i times the sum of the residues:
∫(1 / (z^2 + 1)) dz = 2πi (0 + 0) = 0
In this chapter, we have covered the fundamental concepts and techniques of complex analysis, including complex numbers, complex functions, complex integration, power series, Laurent series, and residue theory. These concepts and techniques provide a solid foundation for further exploration of complex analysis and its applications in various fields.