# Special Functions (theory and applications)

Daniele Ritelli, Giulia Spaletta

• Date:

14 NOVEMBER
-
20 DECEMBER 2023

• Event location: Aula IV, Department of Statistical Sciences, Via Belle Arti 41 (STAT PhD classes virtual room on need)

• Type: Cycle 39 - Mandatory Courses

Aims: Promote knowledge and skills on special mathematical functions often found in statistical problems, and improve analytical thinking and conceptualization skills. To this end, a compact introduction is provided to the scientific environment of Mathematica, widely used in mathematics and statistics; here, the final focus is on special functions.

Learning outcomes: Understanding properties and origin of special functions. The aim is to clarify in the various cases the nature of the problem which gives rise to the need to study a particular function, be it a transcendental equation, a parametric integral, a series or a variable coefficient differential equation;  particular attention is devoted to the analysis of special functions useful in statistical applications. All this is also achieved through the active use of Mathematica. The second module constitutes a hands-on course providing students with the basics to get started: they will learn to work with notebooks, perform symbolic and numeric calculations, generate 2D and 3D graphics, create simple interactive interfaces to visualize and analyze data, and finally elaborate examples on special functions presented in the first module.

Final Exam: A single exam consisting of an individual presentation

Course contents: This course is articulated in two modules.

Module 1: Special Functions: theory and applications (prof. D.Ritelli)
-Lambert W function
-Eulerian functions: Gamma, Beta, Psi and related functions
-Zeta function
-Airy and Hermite
-Bessel functions
-Hypergeometric functions
-Generalized hypergeometric functions
-Incomplete Eulerian functions
-Multivariate hypergeometric functions
-Applications to continuous random variables

Module 2: Mathematica from zero to hero (prof. G.Spaletta)
- Mathematica rules
- Numerical evaluation to arbitrary precision
- Graphics, plots and interactivity
- Using Built-in functions (power, exponential, trigonometric, basic descriptive statistics, etc)
- Defining own's functions
- General solvers for differentiation and integration
- Function expansion in terms of simpler functions
- Function symbolic simplification
- Gamma and related functions