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.

• 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.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

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.

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The in-course element of summative assessment is based on selected tutorial exercises. The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• 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.

Not applicable.

Mathematical Methods in the Physical Sciences, Boas, M. L., Wiley