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Semester 6: B.Sc. Mathematics
Analytic functions and Cauchy-Riemann equations
Analytic functions and Cauchy-Riemann equations
Introduction to Analytic Functions
Analytic functions are complex functions that are differentiable at every point within a given domain. They adhere to the principles of complex differentiability, which is stronger than real differentiability. If a function is analytic, it can be expressed as a power series.
Definition of Cauchy-Riemann Equations
The Cauchy-Riemann equations are a set of two partial differential equations which, if satisfied by the real and imaginary parts of a complex function, ensure that the function is analytic. Specifically, if f(z) = u(x,y) + iv(x,y), then the Cauchy-Riemann equations are: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
Geometric Interpretation
The Cauchy-Riemann equations provide a geometric criteria for analyticity. When both equations are satisfied, it implies that the function preserves angles and is conformal (angle-preserving) in its domain.
Examples of Analytic Functions
Common examples of analytic functions include powers of z (like z^n), exponential functions (e^z), trigonometric functions (sin(z) and cos(z)), and rational functions where the denominator is not zero.
Connection Between Analytic Functions and Cauchy-Riemann Equations
The satisfaction of Cauchy-Riemann equations is both necessary and sufficient for a function to be analytic in a domain. This relationship underscores the importance of these equations in the study of complex analysis.
Applications of Analytic Functions
Analytic functions play a critical role in many areas of mathematics and physics, including fluid dynamics, electromagnetism, and potential theory. Their properties facilitate the simplification of complex phenomena by leveraging the power of complex variable theory.
Conformal mappings and linear fractional transformations
Introduction to Conformal Mappings
Conformal mappings are functions that preserve angles locally. They are essential in complex analysis and have applications in various fields including fluid dynamics and electrostatics.
Properties of Conformal Mappings
These mappings preserve the shape of infinitesimally small figures. They are holomorphic functions with non-zero derivatives, which means they are continuous and differentiable in a particular domain.
Linear Fractional Transformations (LFT)
Linear fractional transformations, also known as Möbius transformations, are a specific type of conformal mapping defined by the formula f(z) = (az + b)/(cz + d) where a, b, c, and d are complex numbers and ad - bc ≠ 0.
Geometric Interpretation of LFT
Geometrically, LFT maps circles and lines in the complex plane to other circles and lines. They preserve the cross-ratio of four points, which is crucial in projective geometry.
Applications of Conformal Mappings and LFT
These transformations are used in various applications including computer graphics, fluid mechanics, and the study of potential flow in physics. They simplify complex problems by transforming them into easier geometrical configurations.
Examples of Conformal Mappings and LFT
Examples include the mapping of the unit disk to the entire complex plane and transformations that convert rectangles into circles. Understanding these examples helps illustrate the theory behind conformal mappings.
Conclusion
Conformal mappings and linear fractional transformations are fundamental concepts in complex analysis that facilitate the understanding and solving of complex problems in multiple domains.
Complex integration and Cauchy integral formula
Complex integration and Cauchy integral formula
Introduction to Complex Integration
Complex integration is the process of integrating functions of complex variables. It is an extension of real integration and has unique properties due to the nature of complex numbers.
Contour Integration
In complex analysis, contour integration involves integrating a function along a path or contour in the complex plane. Contours can be closed or open, and different paths can yield different results based on the properties of the function.
Cauchy's Theorem
Cauchy's theorem states that if a function is holomorphic (analytic) in a simply connected domain, the integral of the function over any closed contour in that domain is zero.
Cauchy Integral Formula
The Cauchy integral formula provides a way to evaluate integrals of holomorphic functions. It states that if f is analytic inside and on some simple closed contour C, then for any point a inside C, the value of f(a) can be expressed as: \[ f(a) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z-a} dz \]
Applications of Cauchy Integral Formula
The Cauchy integral formula is used in evaluating integrals, deriving series expansions, and proving properties of analytic functions. It serves as a foundation for complex function theory.
Residue Theorem
The residue theorem extends the ideas of Cauchy's formula to functions with singularities, allowing for the evaluation of integrals by relating them to the residues of a function at its poles.
Sequences and series of complex functions
Sequences and Series of Complex Functions
Introduction to Sequences of Complex Functions
A sequence of complex functions is a list of functions defined on a domain that converge to a limit function under certain conditions. Understanding sequences is crucial for analyzing the behavior of complex functions.
Convergence of Sequences
A sequence of complex functions {fn} converges to a function f if for every ε > 0, there exists an N such that for all n > N, |fn(z) - f(z)| < ε for all z in the domain. This concept is vital for understanding uniform convergence.
Examples of Convergent Sequences
Consider the sequence fn(z) = z/n. As n approaches infinity, fn(z) converges to 0 for all z in the complex plane. Exploring such examples helps to visualize the concept of convergence.
Series of Complex Functions
A series of complex functions is the sum of an infinite sequence of complex functions. The convergence of series can be absolute or conditional, and it is essential to analyze the convergence behavior.
Tests for Convergence of Series
Various tests like the Weierstrass M-test, ratio test, and root test are utilized to determine the convergence characteristics of series of complex functions.
Power Series and Their Convergence
A power series is a series in the form Σan(z - c)^n, where c is a point in the complex plane. Understanding the radius of convergence and the behavior of power series is fundamental in complex analysis.
Uniform Convergence and Its Implications
Uniform convergence of a sequence or series of complex functions ensures the continuity of the limit function. The distinction between pointwise and uniform convergence is crucial in complex analysis.
Applications of Sequences and Series in Complex Function Theory
Sequences and series play a vital role in the development of other concepts in complex analysis, including analytic functions, residues, and contour integration.
Residues, poles and evaluation of real improper integrals
Residues, poles and evaluation of real improper integrals
Introduction to Residues
Residues are complex numbers associated with singular points of analytic functions. The residue of a function at a pole captures the behavior of the function around that pole.
Types of Poles
Poles are classified into simple poles and higher-order poles. A simple pole is of order one, while a pole of order n has behavior characterized by a specific limit as z approaches the pole.
Residue Theorem
The residue theorem states that if f(z) is a meromorphic function inside and on some simple closed contour C, the integral of f over C is 2πi times the sum of residues of f at its poles inside C.
Calculating Residues
Residues can be calculated using different methods: limit definitions for simple poles, Laurent series expansion, and differentiation for higher-order poles.
Improper Integrals
Improper integrals are integrals that cannot be evaluated using standard techniques due to infinite limits or integrands that do not approach zero.
Evaluation using Residues
For certain improper integrals, residues can be used to evaluate integrals by transforming the integral into a contour integral in the complex plane, allowing the use of the residue theorem.
Example Problems
Illustrative examples that showcase the evaluation of real improper integrals using residues, emphasizing the steps involved in identifying poles, calculating residues, and applying the residue theorem.
