Complex Analysis

Analytic functions are locally given by convergent power series and are infinitely differentiable. Learners will understand the properties of complex-differentiable functions, including conformal mapping, and how to apply them to physical systems.

Cauchy's theorem is a fundamental result in complex analysis concerning contour integrals of holomorphic functions. Learners will understand how to evaluate line integrals around closed paths and apply the residue theorem to solve real integrals.

Complex differentiation extends the concept of the derivative to functions of complex variables. Learners will understand holomorphicity, apply the Cauchy-Riemann equations, and analyze the properties of analytic functions.

Complex integration focuses on integrating complex-valued functions along paths in the complex plane. Learners will master Cauchy's integral theorem and formula, evaluating contour integrals and understanding their applications to real integration.

Complex numbers extend the real number system by introducing the imaginary unit. Learners will understand algebraic operations, polar representation, Euler's formula, and how to solve polynomial equations that have no real roots.

Conformal mapping studies complex functions that preserve angles locally. Learners will understand how to map complicated geometric domains into simpler ones, a technique essential for solving physical problems in fluid dynamics and electrostatics.

Harmonic functions are twice-differentiable real-valued functions that satisfy Laplace's equation. Learners will understand their connection to analytic complex functions, explore the mean value property, and solve boundary value problems in physics.

A Laurent series represents a complex function as a power series containing both positive and negative powers. Learners will analyze functions near singularities, determine poles, and calculate residues for complex integration.

Power series are infinite series of terms with increasing powers of a variable. Learners will determine intervals and radii of convergence, represent analytic functions as Taylor series, and analyze function behavior near specific points.

The residue theorem is a powerful tool in complex analysis for evaluating line integrals of analytic functions. Learners will find residues at singularities and apply the theorem to calculate challenging real and complex integrals.