## MAT 415 Special Functions

### Textbook

### References

- Advanced Engineering Mathematics, Erwin Kreyszig, Wiley
- Advanced Engineering Mathematics, Peter V. O'Neil, WadsWorth
- Mathematical Methods for Physicists, 6th Edition, George B. Arfken, Hans J. Weber, Frank Harris, Academic Press. 7th Edition is available online at ScienceDirect through the library's subscription. 5th Edition hardcopy is available for check-out at the Gulfcoast campus library.
### Topics

### Prerequisites

MAT 280 Multivariable Calculus, MAT 285 Introduction to Differential Equations, MAT 326 Linear Algebra (preferred)
### Class Meeting Times and Class Room

### Office Hours

### Grading

### Lecture Notes

Click on linked topics below to view lecture notes.
#### Modeling a virating drumhead I: an application of Bessel functions

#### Modeling a vibrating drumhead II: an application of Bessel functions

#### Modeling a vibrating drumhead III: an application of Bessel functions

#### Helmhotz Equation

#### Bessel Functions of the First Kind, \(J_{\nu}(x)\) I: Generating Function, Recurrence Relation, Bessel's Equation

#### Example: Cylindrical Resonant Cavity

#### Bessel Functions of the First Kind, \(J_{\nu}(x)\) II: Orthogonality

#### Example: Electrostatic Potential in a Hollow Cylinder

#### Neumann Functions, Bessel Functions of the Second Kind, \(N_{\nu}(x)\)

#### Spherical Bessel Functions

#### Legendre Functions I: A Physical Origin of Legendre Functions

#### Legendre Functions II: Recurrence Relations and Special Properties

#### Legendre functions III: Special Values, Parity, Orthogonality

#### Self-Adjoint Differential Equations I

#### Self-Adjoint Differential Equations II: Hermitian Operators

#### Self-Adjoint Differential Equations III: Orthogonal Eigenfunctions, Gram-Schmidt Orthogonalization

#### An Example: Deuteron

#### Associated Legendre Functions

#### Spherical Harmonics

### Homework Assignments

### Math Forum

If you have a question on your homework or a general math question, you may post it on my Math Forum site. This forum site is equipped with MathJax which supports LaTex, a wonderful type setting system that is really good at formatting mathematical formulas and equations. MathJax allows us to write mathematical formulas and equations in popular web platforms using LaTex code.To include LaTex code in your posts and comments, use:
### Maxima

### Some Useful Internet Resources

- Maxima, A Computer Algebra System
- Maxima CAS Wiki
- Maxima Documentation
- Maxima Guide for Calculus Students
- Maxima Manual
- A Maxima-Gnuplot interface
- Plotting functions in 2d
- Plotting functions in 3d
- Parametric functions
- Implicit functions
- Animations
- Maximal Tutorial
- Maximal Manual
- Newton's Method in Maxima
- Maxima 5.25.0 Manual

There is no particular textbook for the course. Some useful references listed below.

We study some important differential equations that arise in physics and special functions as their solutions. We also discuss some physical applications of special functions. Those special functions include Bessel functions, Legendre functions, Hermite functions, Laguerre functions, Chebyshev polynomials, hypergeometric functions.

Tuesday/Thursday 1:00-2:15pm in SH 309

By appointment.

No tests. The final grade for the course is based on cummutative grades from homework assignments and in-class presentations.

$your-LaTex-code-here$

You can find a list of Latex codes for math symbols here. You may practice your Latexing here.If you want to try Latex in your Windows computer, you may download it for free from MikTex Project Page. If you are using a Linux distribution, usually Latex is a part of the standard packages repository. (The package is called *texlive*.) To use Latex, you need an editor. Texmaker is a nice interface for using Latex. You can edit/compile/view all together in Texmaker. If you don't want to deal with commands, there is a WYSWYG (What You See is What You Get) interface for Latex called Lyx.

In our course we also need to use a mathematics software package for certain computations. We don't have licenses of commercial mathematics software packages such as Maple or Mathematica to use for class, so I have decided to use Maxima which is an open source mathematics software package. The syntax is similar to Mathematica, so those who are familiar with Mathematica can easily adapt to the use of Maxima. To download and install Maxima in your personal computer, go to Maxima Donwload page. Windows and MacOS users will find download files that are suitable for their OS platforms. If you are using Linux distros such as CentOS or Fedora, try to use rpm files available from Maxima download page. Another place you may download Maxima rpm files is here. If you are using Ubuntu, it is really easy to install since Maxima is a part of the standard repository. You can use "Synaptic" to install Maxima or just open a terminal, type "sudo apt-get install maxima" ("sudo apt-get install xmaxima" if you want GUI version) without " " and press enter.