Complex Analysis

Features Complex Analysis

These notes consist of the following chapters in an easy and detailed manner:Chapter 1: Basic Concepts and Complex NumbersChapter 2: Analytic or Regular or Holomorphic FunctionsChapter 3: Elementary Transcendental FunctionsChapter 4: Complex IntegrationChapter 5: Power Series and Related TheoremsChapter 1: Basic Concepts and Complex NumbersIntroduction to Complex NumbersComplex Plane (Argand Diagram)Real and Imaginary PartsComplex ConjugatesModulus (Absolute Value) and ArgumentPolar Form of Complex NumbersOperations on Complex Numbers (Addition, Subtraction, Multiplication, Division)Complex ExponentiationRoots of Complex NumbersComplex Plane GeometryComplex Conjugate and Absolute Value PropertiesEulers FormulaApplications in Engineering and PhysicsChapter 2: Analytic or Regular or Holomorphic FunctionsDefinitions and TerminologyThe Cauchy-Riemann EquationsAnalytic Functions and Holomorphic FunctionsExamples of Analytic FunctionsHarmonic FunctionsConformal MappingMapping Properties of Analytic FunctionsAnalyticity of Elementary FunctionsChapter 3: Elementary Transcendental FunctionsExponential FunctionsLogarithmic FunctionsTrigonometric FunctionsHyperbolic FunctionsInverse Trigonometric and Hyperbolic FunctionsBranch Cuts and Branch PointsAnalytic ContinuationThe Gamma FunctionThe Zeta FunctionChapter 4: Complex IntegrationLine Integrals in the Complex PlanePath Independence and Potential FunctionsContour IntegralsCauchys Integral TheoremCauchys Integral FormulaApplications of Cauchys TheoremMoreras TheoremEstimations of IntegralsChapter 5: Power Series and Related TheoremsPower Series Representation of Analytic FunctionsTaylor Series and Taylors TheoremLaurent SeriesSingularities and the Residue TheoremAnalyticity on the BoundaryApplications of Power SeriesChapter 6: Singularities and Calculus of ResiduesClassification of Singularities (Isolated Singularities, Essential Singularities)Residues and Residue TheoremEvaluation of ResiduesResidue at InfinityApplications of the Residue TheoremPrincipal Value IntegralsChapter 7: Conformal MappingConformal Mappings and their PropertiesMöbius TransformationsConformal Mapping of Simple RegionsConformal Mapping Applications (e.g., solving physical problems)Chapter 8: Contour IntegrationContour Integration TechniquesIntegration Along Real Axis (Jordans Lemma)Residues at PolesCauchys Residue Theorem RevisitedEvaluation of Real Integrals Using Contour IntegrationComplex Integration in Physics and EngineeringChapter 6: Singularities and Calculus of ResiduesChapter 7: Conformal MappingChapter 8: Contour Integration