“A Textbook of Complex Analysis” written by Dipak Kr Ghosh and published by NCBA

Name: "A Textbook of Complex Analysis" written by Dipak Kr Ghosh and published by NCBA - Booksbp.com
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SKU: 9789352551682

1. Foundations of Complex Numbers

Algebraic Properties: Cartesian representation (z = x + iy), fields, conjugation, absolute value, and triangle inequalities.
Geometric Topology: Vector interpretations, representation of lines/planes, limits, open/closed sets, and neighborhoods within the complex plane \(\mathbb{C}\).

2. Analytic Functions & Differentiability

Limits and Continuity: Basic calculus limits mapped onto multi-dimensional inputs.
Analytic Definitions: Holomorphic and entire functions defined via differentiability over an open radius.
Cauchy-Riemann (C-R) Equations: Necessary and sufficient constraints in both Cartesian (x, y) and Polar (r, θ) forms.
Harmonic Functions: Laplace equations and constructing conjugate harmonic functions.

3. Elementary & Multi-valued Functions

Exponential & Trigonometric Extensions: Behavioral properties of \(e^{z}\), \(\sin(z)\), and \(\cos(z)\).
Logarithmic Paradoxes: Principal branches, Riemann surfaces, and managing multivalued solutions.

4. Complex Integration Theory

Contour Integrals: Defining line integrals across smooth piecewise paths.
Cauchy’s Integral Theorem: Proofs showing integrals over closed boundaries evaluate to zero under specific analyticity.
Cauchy’s Integral Formula: Calculating any internal value of a holomorphic function using boundary integrals.
Liouville’s Theorem: Proof that all bounded entire functions are constant, directly leading to the Fundamental Theorem of Algebra.

5. Infinite Series (Taylor & Laurent)

Power Series: Uniform convergence criteria and the radius of convergence.
Taylor Series: Representing smooth analytic functions locally as infinite polynomial series.
Laurent Series: Expanding representation capabilities to account for functional regions surrounding singularities.

6. Residue Calculus & Singularities

Singularity Classification: Isolated singularities, poles, essential singularities, and removable points.
The Residue Theorem: Evaluating difficult real and complex integrals purely via singular point residues.
Improper Integrals: Applying Jordan’s Lemma and contour indentation to calculate definite integrals along the real axis.

7. Conformal Mapping

Geometric Transformation: Preserving angles and scales between mapping spaces.
Möbius (Bilnear) Transformations: Investigating cross-ratios, fixed points, and transformations transforming circles into lines.