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Home/Complex Analysis

Complex Analysis

Rigorous complex-variable theory with proofs, contour integration techniques, residue calculus, conformal maps, and Laplace transform applications for modern engineering, physics, and mathematics.

Higher Education 9 Courses · 45-60 hours Complete Coverage

Courses

Course 1

Complex Numbers Foundations

Available

Master complex number representations, polar form, modulus, argument, and topology of the complex plane including neighborhoods, regions, and connectivity.

Foundation4-5 hours

Topics Covered

Algebraic form

Polar representation

Modulus and argument

Complex plane topology

Course 2

Analytic Functions & Cauchy-Riemann Equations

Available

Explore complex differentiability, Cauchy-Riemann equations, harmonic functions, and the connection between analyticity and differentiability.

Foundation → Intermediate5-6 hours

Topics Covered

Complex limits

Cauchy-Riemann equations

Harmonic functions

Analyticity conditions

Course 3

Elementary Analytic Functions

Available

Study the exponential, logarithmic, trigonometric, hyperbolic, and power functions in the complex domain with their properties and branch cuts.

Intermediate5-6 hours

Topics Covered

Exponential function

Logarithmic function

Trigonometric functions

Power functions

Course 4

Complex Integration & Cauchy's Theorem

Available

Learn contour integration techniques, Cauchy's integral theorem, and methods for evaluating complex integrals along curves.

Intermediate5-6 hours

Topics Covered

Contour integrals

Cauchy's theorem

Path independence

Simply connected domains

Course 5

Cauchy Integral Formula & Derivatives

Available

Master Cauchy's integral formula, derivative estimates, Morera's theorem, and the mean value property of analytic functions.

Intermediate5-6 hours

Topics Covered

Cauchy integral formula

Derivative formulas

Morera's theorem

Mean value property

Course 6

Power Series & Laurent Series

Available

Study power series, Taylor series, Laurent series, convergence criteria, and the classification of isolated singularities.

Intermediate5-6 hours

Topics Covered

Power series

Taylor series

Laurent series

Singularity classification

Course 7

Isolated Singularities & Residue Basics

Available

Understand removable singularities, poles, essential singularities, and learn methods for computing residues.

Intermediate → Advanced5-6 hours

Topics Covered

Singularity types

Residue computation

Pole orders

Residue theorem

Course 8

Residue Applications to Real Integrals

Available

Apply residue theory to evaluate real integrals, trigonometric integrals, Fourier integrals, and improper integrals.

Advanced5-6 hours

Topics Covered

Real integrals

Trigonometric integrals

Fourier integrals

Improper integrals

Course 9

Conformal Mapping & Laplace Transform

Available

Explore conformal mappings, Möbius transformations, and Laplace transform theory with applications to differential equations.

Advanced5-6 hours

Topics Covered

Conformal mappings

Möbius transformations

Laplace transform

Applications to ODEs

Learning Path

Complex

4-5 hours

Analytic

5-6 hours

Elementary

5-6 hours

Complex

5-6 hours

Cauchy

5-6 hours

Power

5-6 hours

Isolated

5-6 hours

Residue

5-6 hours

Conformal

5-6 hours

Ready to Master Complex Variables?

Begin with complex number foundations and advance through analytic functions, contour integrals, residues, and conformal mappings to Laplace transform applications.