In these “Complex AnalysisNotes PDF”, you will study the basic ideas of analysis for complex functions in complex variables with visualization through relevant practicals. Emphasis has been laid on Cauchy’s theorems, series expansions and calculation of residues.
The topics we will cover will be taken from the following list:
Analytic Functions and Cauchy−Riemann Equations: Functions of complex variable, Mappings; Mappings by the exponential function, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives, Differentiation formulae, Cauchy−Riemann equations, Sufficient conditions for differentiability; Analytic functions and their examples.
Elementary Functions and Integrals: 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,
Cauchy’s Theorems and Fundamental Theorem of Algebra: Antiderivatives, Proof of antiderivative theorem, Cauchy−Goursat theorem, Cauchy integral formula; An extension of Cauchy integral formula, Consequences of Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.
Series and Residues: Convergence of sequences and series, Taylor series and its examples; Laurent series and its examples, Absolute and uniform convergence of power series, Uniqueness of series representations of power series, Isolated singular points, Residues, Cauchy’s residue theorem, residue at infinity; Types of isolated singular points, Residues at poles and its examples.
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