Complex Analysis
The complex plane
Maps in C
Complex-valued functions of a complex variable
Complex differentiability
Cauchy–Riemann conditions
Holomorphic functions on an open subset of C
Integration in the complex plane
Cauchy's theorems
Laurent series expansions
Zeros and isolated singularities
Behavior of functions near isolated singularities
Poles
Casorati–Weierstrass theorem for essential singularities
Liouville's theorem
Definition of residue
Residue at infinity
Calculation of residues
Residue theorem
Techniques for evaluating real integrals through analytic continuation into C
Jordan's lemma
Analytic continuation and multivalued functions

Functional Spaces
Review of topological spaces, metric spaces, and Banach spaces
Sequence spaces lp and l∞
Hilbert spaces
Inner products and orthogonality
Complete orthonormal systems
Fischer–Riesz theorem
Lp and L∞
Important orthonormal systems:
Fourier series
Hermite polynomials
Legendre polynomials
Laguerre polynomials
Bessel functions

Linear Operators
Linear operators on Hilbert spaces and their properties
Continuous and bounded operators
Operator norm
Spectral problems and classification of eigenvalues
Definition of eigenfunctions
Self-adjoint operators
Eigenvalues and eigenfunctions of self-adjoint operators
Spectral decomposition theorem
Projectors and their properties
Isometric operators and their properties
Unitary operators and their properties
Linear functionals on Hilbert spaces
Dual space
Riesz representation theorem

Fourier Transform
Definition on L1 and L2 and main properties
Techniques for computing Fourier transforms
Applications to solving differential equations relevant to physics

Laplace Transform
Definition and main properties
Inverse transform
Techniques for computing Laplace transforms
Applications to solving differential equations relevant to physics