BSc (NEP) - VI Sem - Complex Analysis

 

COMPLEX ANALYSIS

Semester 6th

A Comprehensive Guide

B.Sc. Mathematics - VI, PAPER 1- Part B

COMPLEX ANALYSIS

Complex Analysis: An Introduction

1. Introduction to Complex Numbers Complex analysis is the study of functions that operate on complex numbers. A complex number is of the form:

$$z = x + iy$$

where $x$ and $y$ are real numbers, and $i$ is the imaginary unit satisfying $i^2 = -1$.

The modulus of $z$ is given by:

$$|z| = \sqrt{x^2 + y^2}$$

and the argument (angle) of $z$ is:

$$\arg(z) = \tan^{-1}(y/x)$$

2. Analytic Functions A function $f(z)$ is said to be analytic at a point if it has a derivative at every point in a neighborhood of that point. For a function to be analytic, it must satisfy the Cauchy-Riemann equations:

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

where $f(z) = u(x,y) + iv(x,y)$ and $u, v$ are real-valued functions.

3. Complex Differentiation and Integration The derivative of a complex function is defined similarly to real functions:

$$f'(z) = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}$$

For integration, a contour (or path) integral is defined along a curve $C$:

$$\int_C f(z) \, dz$$

A fundamental result in complex analysis is Cauchy's Integral Theorem, which states that if a function is analytic in a simply connected domain, then:

$$\oint_C f(z) \, dz = 0$$

for any closed contour $C$ in the domain.

4. Taylor and Laurent Series A complex function can be expanded as a power series. If $f(z)$ is analytic, it can be expressed as a Taylor series:

$$f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$$

where $a_n$ are the coefficients.

For functions with singularities, the Laurent series is used, which includes negative power terms:

$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$$

5. Residue Theorem and Applications Residues are key in evaluating complex integrals. The Residue Theorem states that for a function with isolated singularities inside a contour $C$:

$$\oint_C f(z) \, dz = 2\pi i \sum \text{Residues at singularities}$$

This theorem is useful in evaluating real integrals and solving differential equations.

  Syllabus

(NEP SYLLABUS MODIFIED BY CCS UNIVERSITY)

Complex Analysis

B.Sc. MATHEMATICS SEMESTER-VI Paper 1-Part B

AS PER NEP 2020 [NATIONAL EDUCATION POLICY]

B.A./B.Sc. Paper-I : Part B

Unit I

Functions of complex variable, Mappings; Mappings by the exponential function, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives, Differentiation formulae.

Unit II

Analytic Functions Cauchy-Riemann equations, Sufficient conditions for differentiability; Analytic functions and their examples, Harmonic function Method of constructing a regular function (Milne-Thomson’s method).

Unit III

Conformal mapping, necessary and sufficient condition, Inverse point, Bilinear transformation, critical point, cross ratio, fixed point.

Unit IV

Exponential function, Logarithmic function, Branches and derivatives of logarithms, Trigonometric function, Derivatives of functions, Definite integrals of functions, Contours, Contour integrals and its examples, Upper bounds for moduli of contour integrals.

Book Chapters Name

Chapter 1- Complex Numbers and Their Geometrical Representation (Solution)

Chapter 2- Analytic Functions (Coming Soon)

Chapter 3- Comformal Mappings (Coming Soon)

Chapter 4- More About Conformal Mappings (Mapping by Elementary Functions)(Coming Soon)

Chapter 5- Power Series and Elementary Functions(Coming Soon)

Chapter 6- Complex Integration(Coming Soon)