MA2004: Series and Transforms
| School | Cardiff School of Mathematics |
| Department Code | MATHS |
| Module Code | MA2004 |
| External Subject Code | 100405 |
| Number of Credits | 10 |
| Level | L5 |
| Language of Delivery | English |
| Module Leader | Professor Nicolas Dirr |
| Semester | Spring Semester |
| Academic Year | 2022/3 |
Outline Description of Module
A lecture based module, which deals with fundamental mathematical methods which are essential to all students of mathematics or statistics. In particular the theory of certain important series and transforms is developed.
On completion of the module a student should be able to
- Understand what a Fourier Series is, and obtain series in certain cases.
- Know how to find the Laplace Transform of basic functions, and find inverse transforms. Be able to use Laplace Transforms to solve differential equations.
- Recognise and manipulate Legendre Polynomials and Bessel Functions.
- Find series solutions for certain differential equations.
How the module will be delivered
Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:
- Weekly face to face classes (e.g. labs, lectures, exercise classes)
- Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)
Students are also expected to undertake self-guided study throughout the duration of the module.
Skills that will be practised and developed
Skills:
Expand periodic functions in Fourier series, solve differential equations using Laplace Transforms, and obtain series solutions where appropriate, manipulate certain orthogonal functions.
Transferable Skills:
Recognition of type of solution possible for a differential equation, algebraic manipulation.
Assessment Breakdown
| Type | % | Title | Duration(hrs) |
|---|---|---|---|
| Exam - Spring Semester | 100 | Series And Transforms | 2 |
Syllabus content
- Fourier series
- Periodic and orthogonality properties of cosine and sine functions.
- Derivation of series for function of period 2p
- Odd and even functions and their series.
- Half-range series.
- Functions of period 2l
- Laplace Transforms
- Definition and derivation of transforms of elementary functions.
- Shift theorems.
- Inversion using partial fraction and recognition.
- Applications to ordinary differential equations.
- Orthogonal Functions
- Legendre polynomials, derivation, recurrence relations, generating function, orthogonal integrals, Rodrigues’ Formula.
- Introduction to Bessel functions, recurrence relations, generating functions, orthogonal integrals.
- Series solution of differential equations
- The method of Frobenius.
- Solutions at non-singular points
- Singular points and solution at regular singular points
- Solutions when the roots of the indicial equation do not differ by an integer or zero.
- The solution of some types when the roots do differ by an integer.
- Brief discussion of equations whose roots differ by an integer.