Undergraduate Course: Fourier Analysis and Statistics (PHYS09055)
Course Outline
School  School of Physics and Astronomy 
College  College of Science and Engineering 
Course type  Standard 
Availability  Available to all students 
Credit level (Normal year taken)  SCQF Level 9 (Year 3 Undergraduate) 
Credits  20 
Home subject area  Undergraduate (School of Physics and Astronomy) 
Other subject area  None 
Course website 
None 
Taught in Gaelic?  No 
Course description  Full details to follow. It will be seen as a coherent 20pt course by all single honours physics students, with one three hour exam in the December diet. For joint honours it needs to be available as two 10pt halves, so that they can drop one half. For the half courses there are two hour exams in the December diet. 
Entry Requirements (not applicable to Visiting Students)
Prerequisites 

Corequisites  
Prohibited Combinations  Students MUST NOT also be taking
Fourier Analysis (PHYS09054)

Other requirements  None 
Additional Costs  None 
Information for Visiting Students
Prerequisites  None 
Displayed in Visiting Students Prospectus?  No 
Course Delivery Information

Delivery period: 2014/15 Semester 1, Available to all students (SV1)

Learn enabled: No 
Quota: None 

Web Timetable 
Web Timetable 
Course Start Date 
15/09/2014 
Breakdown of Learning and Teaching activities (Further Info) 
Total Hours:
200
(
Lecture Hours 22,
Seminar/Tutorial Hours 22,
Formative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
149 )

Additional Notes 

Breakdown of Assessment Methods (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)  Fourier Analysis and Statistics  3:00  
Summary of Intended Learning Outcomes
 State in precise terms key concepts relating to Fourier analysis, the solution of partial differential equations and probability and statistics.
 Apply standard methods of Fourier analysis, the solution of partial differential equations and probability and statistics to solve physical problems of moderate complexity.
 Justify the logical steps in an extended solution of a physics problem.
 Use experience and intuition gained from solving physics problems to predict the likely range of reasonable solutions to an unseen problem.
 Resolve conceptual and technical difficulties by locating and integrating relevant information from a diverse range of sources (including computer algebra packages where appropriate).

Assessment Information
Coursework 20% and examination 80%. 
Special Arrangements
None 
Additional Information
Academic description 
Not entered 
Syllabus 
Fourier Analysis (20 lectures)
 Linear algebra view of functions: orthonormal basis set, expansion in series
 Fourier series: sin and cos as a basis set; calculating coefficients; examples of waves; Complex functions; convergence, Gibbs phenomenon.
 Fourier transform; uncertainty principle.
 Solving Ordinary Differential Equations with Fourier methods.
 Applications of Fourier transforms: Fraunhofer diffraction; Quantum scattering; forceddamped oscillators; wave equation; diffusion equation.
 Alternative methods for wave equations: d'Alembert's method; extension to nonlinear wave equation.
 Convolution; Correlations; Parseval's theorem.
 Power spectrum; Sampling; Nyquist theorem.
 Dirac delta; Fourier representation.
 Green's functions for 2nd order ODEs.
 SturmLiouville theory: orthogonality and completeness.
Partial Differential Equations, Probability & Statistics (20 lectures)
 Overview of Partial Differential Equations in Physics: Poisson, Wave, Diffusion, Continuity, Laplace, Schrodinger.
 Separation of variables.
 Examples with rectangular symmetry: 'rectangular harmonics'.
 Examples with circular symmetry: Bessel functions.
 Examples with spherical symmetry: Legendre polynomials, spherical harmonics; Charged sphere, gravitational potential.
 Probability of discrete events; Multiple events: joint, conditional and marginal distributions.
 Bayes' theorem; frequentist view; probability as a degree of belief.
 Generalisation of probability to continuous variables.
 Permutations, combinations.
 Random walk and the binomial distribution; Stirling's approximation; Gaussian and Poisson distributions as limiting cases.
 Functions of a random variable: expectations, moments; Fourier transform of probability distribution as moment generating function; Application to sampling nonuniform random numbers.
 Addition of random variables as a convolution; addition of Gaussian distributions; centrallimit theorem.
 Estimating mean, variance, error on the mean from finite data sets.
 Cumulative distribution and centiles; error function; hypothesis testing, confidence limits.
 Least squares fitting; goodness of fit; x2 distribution; maximum likelihood; improbably good and poor fits.
 Residuals; error analysis; KolmogorovSmirnov test.
 Linear regression; correlations.

Transferable skills 
Not entered 
Reading list 
Not entered 
Study Abroad 
Not entered 
Study Pattern 
Not entered 
Keywords  FASt 
Contacts
Course organiser  Prof John Peacock
Tel: (0131) 668 8390
Email: John.Peacock@ed.ac.uk 
Course secretary  Mrs Bonnie Macmillan
Tel: (0131 6)50 5905
Email: Bonnie.MacMillan@ed.ac.uk 

