Learn complex analysis: analytic functions, Cauchy-Riemann and residues
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Sub Topics
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MCQs
35
MCOs
61
True/False
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Fill Blanks
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Rearrange
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Matching
18
Comprehensions
29
Flashcard Decks
Curriculum
What You'll Learn
01 Complex Numbers and Functions 3 topics
1 The Complex Plane
- Cartesian and Polar Forms
- Modulus and Argument
- Complex Conjugation
- Regions in the Complex Plane
2 Complex Functions
- Functions of a Complex Variable
- Limits and Continuity
- Differentiability
- The Cauchy-Riemann Equations
3 Elementary Complex Functions
- Polynomials and Rational Functions
- Exponential and Logarithmic Functions
- Trigonometric and Hyperbolic Functions
- Complex Powers and Roots
02 Analytic Functions 3 topics
1 Holomorphic Functions
- Definition and Properties
- Harmonic Functions
- Conformal Mappings
2 Power Series
- Convergence of Series
- Radius of Convergence
- Term-by-Term Differentiation and Integration
- Taylor Series
- Laurent Series
3 Singularities and Zeros
- Removable Singularities
- Poles
- Essential Singularities
- Classification of Singularities
- Zeros and Their Order
03 Complex Integration 3 topics
1 Complex Line Integrals
- Definition and Properties
- Independence of Path
- Antiderivatives
2 Cauchy's Theorem
- Simply and Multiply Connected Domains
- Cauchy-Goursat Theorem
- Deformation of Contours
- Cauchy's Integral Formula
- Extensions of Cauchy's Integral Formula
3 Applications of Cauchy's Theorem
- Liouville's Theorem
- Maximum Modulus Principle
- Fundamental Theorem of Algebra
- Morera's Theorem
04 Residue Theory 2 topics
1 The Residue Theorem
- Definition of Residues
- Calculation of Residues
- Cauchy's Residue Theorem
2 Applications of Residue Theory
- Evaluation of Definite Integrals
- Improper Integrals
- Integration of Trigonometric Functions
- Jordan's Lemma
- Indented Contours
05 Conformal Mapping 3 topics
1 Geometric Properties
- Preservation of Angles
- Local and Global Behavior
- Fixed Points
2 Elementary Mappings
- Linear Fractional Transformations
- The Riemann Sphere
- Cross-Ratio
- Schwarz-Christoffel Transformations
3 Applications of Conformal Mapping
- Fluid Flow
- Electrostatics
- Heat Conduction
- Boundary Value Problems
06 Special Functions and Methods 3 topics
1 Gamma and Beta Functions
- Definitions and Properties
- Reflection and Duplication Formulas
- Asymptotic Behavior
2 Elliptic Functions
- Doubly Periodic Functions
- Weierstrass Functions
- Jacobian Elliptic Functions
3 Riemann Zeta Function
- Definition and Basic Properties
- Functional Equation
- Zeros and the Riemann Hypothesis
07 Advanced Topics 4 topics
1 Entire Functions
- Order and Type
- Hadamard's Factorization Theorem
- Picard's Theorems
2 Analytic Continuation
- Principle of Analytic Continuation
- Schwarz Reflection Principle
- Riemann Surfaces
3 Infinite Products
- Convergence Criteria
- Weierstrass Products
- Mittag-Leffler Theorem
4 Normal Families
- Montel's Theorem
- Equicontinuity
- Arzela-Ascoli Theorem
08 Applications in Physics and Engineering 4 topics
1 Potential Theory
- Laplace's Equation
- Dirichlet and Neumann Problems
- Green's Functions
2 Signal Processing
- Fourier and Laplace Transforms
- Transfer Functions
- Stability Analysis
3 Fluid Dynamics
- Complex Potential
- Source, Sink, and Vortex Flows
- Airfoil Theory
4 Quantum Mechanics
- Schrödinger Equation in Complex Form
- Path Integrals
- Analytic Properties of Wave Functions
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Complex Analysis
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