Università Roma Tre

Complex numbers:
Algebraic operations; Cartesian, trigonometric and exponential forms; Powers and roots; Equations.

Taylor expansion:
Taylor formula; Taylor expansion of the elementary functions; Operations on the Taylor expansion.

Numerical series:
Numerical sequences; numerical series; positive term series; alternate sign term series; algebraic operations on series.

Fourier series:
Trigonometric polynomials; Fourier series and coefficients; Exponential form of the Fourier series; Derivation of the Fourier series; Convergence of the Fourier series; Periodic functions with period T0.

Fourier Transform:
Dirac delta function; Introduction to the Fraunhofer diffraction; Definition of the Fourier transform and antitransform; Examples of Fourier transforms; Mathematical properties of the Fourier transform; Physical properties of the Fourier transform; Multidimension transforms; Spatial filter.

Zernike Polynomials

Ordinary differential equations:
General definitions; First order equations; Second order linear differential equations with constant coefficients.

Claudio Canuto, Anita Tabacco ``Analisi Matematica I" [C.T.I]
Claudio Canuto, Anita Tabacco ``Analisi Matematica II" [C.T.II]
Greg Gbur ``Mathematical Methods for Optical Physics and Engineering" [Gbur]

1- Complex numbers; Taylor developments and applications.
2- Numerical series: References on sequences; Numerical series; Series with positive terms; Series with terms of alternate sign; Algebraic operations on the series.
3- Fourier series: Trigonometric polynomials; Fourier coefficients and series; Exponential form of the Fourier series; Fourier series and derivation;
Convergence of the Fourier series; Periodic functions of period T 0.
4- Fourier transform: Introduction to Fraunhofer diffraction; Definition of Fourier transforms and anti-transforms; Examples of Fourier transforms; Mathematical properties of the Fourier transform; Physical properties of the Fourier transforms; Self-functions of the Fourier operator; Fourier transform in multidimensional spaces; Spatial filter.
5- Zernike polynomials
6- Ordinary differential equations: General definitions; First order equations; Examples; Scalar equations of the first
order; The Cauchy problem for the equations of the first order, Linear equations of the second order with constant coefficients.

Claudio Canuto, Anita Tabacco, "Analisi Matematica I"

Claudio Canuto, Anita Tabacco, "Analisi Matematica II"

Greg Gbur, "Mathematical Methods for Optical Physics and Engineering"