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# Learn Complex Variables Theory and Applications with Kasana's Book

## Complex Variables Theory and Applications Kasana PDF

If you are interested in learning about complex variables theory and applications, you might have come across a book by H.S. Kasana titled Complex Variables: Theory And Applications. This book is one of the popular textbooks on this subject that covers both the basics and the advanced topics. But is this book worth reading? What does it offer to the readers? How does it compare to other books on complex variables? In this article, we will answer these questions by giving you an overview of complex variables theory and applications kasana pdf.

## What are complex variables?

Before we dive into the book review, let us first understand what complex variables are. Complex variables are numbers that have both real and imaginary parts. For example, z = 3 + 4i is a complex variable where 3 is the real part and 4i is the imaginary part. The imaginary part is denoted by i which is defined as i = -1. The set of all complex numbers is denoted by C.

Complex numbers can be represented in different ways. One way is to use the rectangular or Cartesian form where z = x + yi where x and y are real numbers. Another way is to use the polar or trigonometric form where z = r(cosθ + i sinθ) where r is the modulus or absolute value of z and θ is the argument or angle of z. The modulus r can be found by using Pythagoras' theorem as r = (x^2 + y^2) while the argument θ can be found by using trigonometry as θ = tan^-1(y/x). The polar form can also be written using Euler's formula as z = re^iθ.

Complex numbers have many properties and operations that are similar to real numbers, such as addition, subtraction, multiplication, division, equality, inequality, etc. However, complex numbers also have some properties and operations that are different from real numbers, such as conjugation, modulus, argument, roots of unity, etc. For example, the conjugate of a complex number z = x + yi is defined as z* = x - yi which is obtained by changing the sign of the imaginary part. The modulus of a complex number z = x + yi is defined as z = (x^2 + y^2) which is the distance from z to the origin in the complex plane. The argument of a complex number z = x + yi is defined as arg(z) = θ where θ is the angle from the positive x-axis to the line joining z and the origin in the complex plane.

## Why are complex variables important?

Complex variables are not only fascinating mathematical objects but also powerful tools for solving many problems in mathematics, physics, engineering, and other fields. Complex variables can help us understand and analyze phenomena that involve waves, oscillations, vibrations, heat transfer, fluid dynamics, electromagnetism, quantum mechanics, relativity, cryptography, signal processing, control theory, and more. Complex variables can also help us simplify and generalize many results and methods that are otherwise difficult or impossible to obtain using real variables. For example, complex variables can help us solve differential equations, evaluate integrals, find series expansions, perform transformations, etc.

One of the reasons why complex variables are so useful and versatile is that they have a rich and elegant theory that connects various concepts and techniques in a coherent and consistent way. Complex variables theory is based on the notion of complex functions which are functions that map complex numbers to complex numbers. Complex functions have many remarkable properties and behaviors that make them suitable for modeling and solving many problems. For instance, complex functions can be continuous, differentiable, integrable, analytic, harmonic, etc. Complex functions can also be represented by power series, Laurent series, Fourier series, etc. Complex functions can also be integrated along curves or contours using Cauchy's integral theorem and formula. Complex functions can also have singularities or points where they are undefined or infinite. Complex functions can also be mapped from one region to another using conformal mapping which preserves angles and shapes.

## How to learn complex variables theory?

Complex variables theory is a vast and deep subject that requires a solid background in calculus and linear algebra. However, it is also a rewarding and enjoyable subject that offers many insights and applications. To learn complex variables theory effectively, one needs to master the main topics and concepts that form the core of the subject. Some of these topics and concepts are:

### Complex functions

A complex function is a function that maps complex numbers to complex numbers. For example, f(z) = z^2 + 1 is a complex function where z is a complex variable. A complex function can be written in terms of its real and imaginary parts as f(z) = u(x,y) + iv(x,y) where u(x,y) and v(x,y) are real functions of two real variables x and y. A complex function can also be written in terms of its modulus and argument as f(z) = f(z)e^iarg(f(z)) where f(z) and arg(f(z)) are real functions of z.

A complex function can have a domain which is the set of all complex numbers for which the function is defined. For example, the domain of f(z) = 1/z is C - 0 which is the set of all complex numbers except zero. A complex function can also have a range which is the set of all complex numbers that are mapped by the function. For example, the range of f(z) = 1/z is C - 0 which is the same as its domain.

A complex function can be graphed in different ways. One way is to use the complex plane where each point represents a complex number z = x + yi with x as the horizontal axis and y as the vertical axis. Another way is to use two real planes where one plane represents the real part u(x,y) of f(z) and another plane represents the imaginary part v(x,y) of f(z). A third way is to use a three-dimensional space where each point represents a pair (z,f(z)) with z as the horizontal plane and f(z) as the vertical axis.

### A complex function can have various operations such as addition, subtraction, multiplication, division, composition, inversion, etc. For example, - Addition: (f + g)(z) = f(z) + g(z) - Multiplication: (f * g)(z) = f(z) * g(z) - Division: (f / g)(z) = f(z) / g(z) if g(z) 0 - Composition: (f g)(z) = f(g(z)) - Inversion: f^-1(z) = w such that f(w) = z if f is one-to-one Analyticity

A complex function is said to be analytic or holomorphic at a point z0 if it is differentiable at z0 and at every point in some neighborhood of z0. A complex function is said to be analytic or holomorphic in a region D if it is analytic at every point in D. A complex function is said to be entire if it is analytic in the whole complex plane C.

Analytic functions have many remarkable properties and behaviors that make them suitable for modeling and solving many problems. Some of these properties and behaviors are: - Derivatives: An analytic function has derivatives of all orders at every point in its domain. The derivative of an analytic function f(z) at a point z0 is defined as f'(z0) = lim_(zz0)(f(z) - f(z0))/(z - z0) which is the slope of the tangent line to the graph of f(z) at z0. The derivative of an analytic function can also be found by using the Cauchy-Riemann equations which are u_x = v_y and u_y = -v_x where u and v are the real and imaginary parts of f(z). - Integrals: An analytic function can be integrated along any curve or contour C in its domain. The integral of an analytic function f(z) along a contour C is defined as _C f(z) dz which is the sum of the products of f(z) and the infinitesimal changes in z along C. The integral of an analytic function can also be evaluated by using Cauchy's integral theorem and formula which state that if f(z) is analytic in a simply connected region D and C is a closed contour in D, then _C f(z) dz = 0 and f(z0) = (1/2πi)_C (f(z))/(z - z0) dz for any z0 in D. - Series: An analytic function can be represented by a power series or a Laurent series in some region around a point z0. A power series is a series of the form _(n=0)^ a_n (z - z0)^n where a_n are complex coefficients. A Laurent series is a series of the form _(n=-)^ a_n (z - z0)^n where a_n are complex coefficients. A power series converges absolutely and uniformly in a disk z - z0 1 is called a multiple zero. An analytic function can have only finitely many zeros in a bounded region unless it is identically zero.

### Cauchy-Riemann equations

The Cauchy-Riemann equations are two partial differential equations that relate the real and imaginary parts of an analytic function. The Cauchy-Riemann equations are u_x = v_y and u_y = -v_x where u and v are the real and imaginary parts of f(z). The Cauchy-Riemann equations can be derived by equating the real and imaginary parts of the definition of the derivative of an analytic function.

The Cauchy-Riemann equations have many applications and implications in complex variables theory. Some of these applications and implications are: - Analyticity: The Cauchy-Riemann equations are necessary and sufficient conditions for a complex function to be analytic in a region D if the function and its partial derivatives are continuous in D. The Cauchy-Riemann equations can also be used to test whether a given function is analytic or not at a point or in a region. - Derivatives: The Cauchy-Riemann equations can be used to find the derivative of an analytic function f(z) at a point z0 by using the formula f'(z0) = u_x(z0) + iv_x(z0) where u and v are the real and imaginary parts of f(z). The Cauchy-Riemann equations can also be used to find higher order derivatives of an analytic function by using the chain rule and the fact that the partial derivatives of u and v also satisfy the Cauchy-Riemann equations. - Harmonicity: The Cauchy-Riemann equations imply that the real and imaginary parts of an analytic function are harmonic functions, which means that they satisfy Laplace's equation ^2u = 0 and ^2v = 0 where ^2 is the Laplacian operator. Harmonic functions have many properties and applications in physics and engineering, such as being solutions of heat equation, wave equation, potential theory, etc.

### Contour integration

Contour integration is a technique of integrating a complex function along a curve or contour in the complex plane. A contour is a continuous curve that can be smooth or piecewise smooth. A contour can also be closed or open. A closed contour is a curve that ends at the same point where it starts. An open contour is a curve that does not end at the same point where it starts.

Contour integration can be performed by using the definition of the integral of a complex function along a contour C as _C f(z) dz = lim_(n) _(k=1)^n f(z_k)(z_k - z_k-1) where z_k are points on C that divide C into n subintervals and z_k - z_k-1 are the infinitesimal changes in z along C. Contour integration can also be performed by using parametrization which is a method of representing a contour C by a function z(t) where t is a real parameter that varies from t1 to t2. In this case, the integral of a complex function along C can be written as _C f(z) dz = _(t1)^(t2) f(z(t))z'(t) dt where z'(t) is the derivative of z(t) with respect to t.

Contour integration has many applications and advantages in complex variables theory. Some of these applications and advantages are: - Cauchy's integral theorem: This is a fundamental result that states that if f(z) is analytic in a simply connected region D and C is a closed contour in D, then _C f(z) dz = 0. This means that the integral of an analytic function along a closed contour does not depend on the shape or orientation of the contour but only on the endpoints of the contour. This also means that any two contours with the same endpoints in D have the same integral value. - Cauchy's integral formula: This is another fundamental result that states that if f(z) is analytic in a simply connected region D and C is a closed contour in D, then f(z0) = (1/2πi)_C (f(z))/(z - z0) dz for any z0 in D. This means that the value of an analytic function at any point inside a closed contour can be found by using the values of the function on the contour. This also means that an analytic function is completely determined by its values on any closed contour that encloses its domain. - Residue theorem: This is an important result that states that if f(z) is analytic in a region D except for isolated singularities z1, z2, ..., zn and C is a closed contour in D that encloses these singularities, then _C f(z) dz = 2πi_(k=1)^n Res(f,z_k) where Res(f,z_k) is the residue of f(z) at z_k which is defined as Res(f,z_k) = lim_(zz_k)(z - z_k)f(z). This means that the integral of a complex function along a closed contour can be found by using the residues or coefficients of (z - z_k)^-1 in the Laurent series expansion of f(z) at each singularity. This also means that many complex integrals can be simplified or evaluated by using residues.

### Residues

where it is undefined or infinite. A residue can be defined as the coefficient of (z - z0)^-1 in the Laurent series expansion of a complex function f(z) at a singularity z0. A residue can also be defined as the limit of (z - z0)f(z) as z approaches z0. A residue can also be defined as the integral of f(z) along a small circle around z0 divided by 2πi.

Residues have many applications and uses in complex variables theory. Some of these applications and uses are: - Classification of singularities: A singularity of a complex function f(z) at a point z0 can be classified into different types based on the value and existence of the residue of f(z) at z0. A singularity is said to be removable if Res(f,z0) = 0 which means that f(z) can be extended to a continuous and analytic function at z0 by defining f(z0) = lim_(zz0)f(z). A singularity is said to be a pole of order m if Res(f,z0) 0 and lim_(zz0)(z - z0)^m+1f(z) = 0 for some positive integer m which means that f(z) behaves like (z - z0)^-m near z0. A singularity is said to be essential if Res(f,z0) 0 and lim_(zz0)(z - z0)^m+1f(z) 0 for any positive integer m which means that f(z) behaves erratically near z0. - Evaluation of residues: The residue of a complex function f(z) at a singularity z0 can be evaluated by using different methods depending on the type and nature of the singularity and the function. Some of these methods are: - Laurent series method: This method involves finding the Laurent series expansion of f(z) at z0 and identifying the coefficient of (z - z0)^-1 as the residue. This method is applicable for any type of singularity but it may be difficult or tedious to find the Laurent series in some cases. - Limit method: This method involves finding the limit of (z - z0)f(z) as z approaches z0 as the residue. This method is applicable for simple poles or poles of order 1 but it may not work for higher order poles or essential singularities. - Cauchy's integral formula method: This method involves finding the integral of f(z) along a small circle around z0 divided by 2πi as the residue. This method is applicable for any type of singularity but it may require contour integration techniques to evaluate the integral in some cases. - Residue theorem: The residue theorem is an important result that states that if f(z) is analytic in a region D except for isolated singularities z1, z2, ..., zn and C is a closed contour in D that encloses these singularities, then _C f(z) dz = 2πi_(k=1)^n Res(f,z_k). The residue theorem can be used to simplify or evaluate many complex integrals by using residues instead of contour integration techniques.

### Conformal mapping

Conformal mapping is a technique of mapping or transforming one region in the complex plane to another region by using a complex function. A conformal mapping is a mapping that preserves angles and shapes locally, which means that if two curves intersect at an angle α in one region, then their images under the mapping also intersect at an angle α in another region. A conformal mapping is also one-to-one and onto, which means that it maps each point in one region to a unique point in another region and vice versa.

Conformal mapping has many applications and benefits in complex variables theory. Some of these applications and benefits are: - Solving boundary value problems: Boundary value problems are problems that involve finding an unknown function that satisfies a given differential equation and some boundary conditions in a region. Conformal mapping can help solve boundary value problems by transforming a complicated region into a simpler region where the solution can be found more easily. For example, conformal mapping can help solve Laplace's equation ^2u = 0 in a region D by mapping D to a disk where the solution can be found by using harmonic functions. - Finding analytic functions: Analytic functions are functions that are differentiable at every point in their domain. Conformal mapping can help find analytic functions by mapping a region D to another region where an analytic function is known and then applying the inverse mapping to obtain an analytic function in D. For example, conformal mapping can help find an analytic function in a half-plane by mapping it to a disk where an analytic function can be found by using power series. - Visualizing complex functions: Complex functions are functions that map complex numbers to complex numbers. Conformal mapping can help visualize complex functions by mapping a region D to another region where the image of the function can be seen more clearly. For example, conformal mapping can help visualize a complex function in a disk by mapping it to a strip where the image of the function can be seen as a color map.

## What is Kasana's book about?

Kasana's book on complex variables theory and applications is one of the popular textbooks on this subject that covers both the basics and the advanced topics. The book is divided into 14 chapters and 4 appendices that span over 600 pages. The book is intended for undergraduate and graduate students of mathematics, physics, engineering, and other fields that require complex variables. The book is also suitable for self-study and reference for researchers and practitioners.

The book aims to provide a comprehensive and rigorous treatment of complex variables theory and applications with an emphasis on clarity, comprehensiveness, examples, exercises, and features. The book also aims to strike a balance between theory and practice, abstraction and concreteness, generality and specificity, and simplicity and complexity. The book also aims to expose the readers to the beauty and elegance of complex variables theory and applications and to stimulate their interest and curiosity.

### Overview of the book

The book consists of 14 chapters and 4 appendices that cover the following topics: - Chapter 1: Introduction. This chapter introduces the basic concepts and properties of complex numbers, such as algebraic operations, geometric representation, polar form, Euler's formula, roots of unity, etc. This chapter also introduces some elementary functions of complex variables, such as exponential function, logarithmic fu

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