Undergraduate Course: Fourier Analysis and Statistics (PHYS09055)
Course Outline
School  School of Physics and Astronomy 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 9 (Year 3 Undergraduate) 
Availability  Available to all students 
SCQF Credits  20 
ECTS Credits  10 
Summary  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. 
Course description 
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.

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 
Course Delivery Information

Academic year 2014/15, Available to all students (SV1)

Quota: None 
Course Start 
Semester 1 
Timetable 
Timetable 
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 )

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

Additional Information (Assessment) 
Coursework 20% and examination 80%. 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)  Fourier Analysis and Statistics  3:00  
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).

Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  FASt 
Contacts
Course organiser  Prof John Peacock
Tel: (0131) 668 8390
Email: John.Peacock@ed.ac.uk 
Course secretary  Yuhua Lei
Tel: (0131 6) 517067
Email: yuhua.lei@ed.ac.uk 

