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Programme & Unit Catalogues

## PH30025: Mathematical methods

Owning Department/School: Department of Physics
Credits: 6
Level: Honours (FHEQ level 6)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: Mandatory extra work (where allowed by programme regulations)
Requisites: Before taking this unit you must take PH20019 and take PH20020
Description: Aims:
The aim of this unit is to continue the development of students' mathematical knowledge and skills by introducing concepts and methods used in a mathematical description of the physical world.

Learning Outcomes:
After taking this unit the student should be able to:
* derive theorems of analytic functions and use them to evaluate integrals;
* use superposition methods for inhomogeneous equations;
* solve problems in scattering theory;
* derive the Euler-Lagrange equation and solve problems using the calculus of variations.

Skills:
Numeracy T/F A, Problem Solving T/F A.

Content:
Functions of a complex variable (10 hours): Functions of z, multivalued functions, branch points and branch cuts. Differentiation, analytic functions, Cauchy-Riemann equations. Complex integration; Cauchy's theorem and integral. Taylor and Laurent expansions. Residue theorem, evaluation of real integrals. Kramers-Kronig relations.
Superposition methods (8 hours): Sturm-Liouville theory, eigenfunctions and eigenvalues, orthogonality of eigenfunctions. Solution of inhomogeneous equations. Green's functions. Examples using 1D and 3D operators. Scattering theory.
Calculus of variations (4 hours): Euler-Lagrange equation: derivation and examples. Inclusion of constraints: Lagrange multipliers, examples.
Programme availability:

#### PH30025 is Optional on the following programmes:

Programmes in Natural Sciences
• UXXX-AFB01 : BSc (hons) Natural Sciences (Full-time) - Year 3
• UXXX-AKB02 : BSc (hons) Natural Sciences with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 4
• UXXX-AAB02 : BSc (hons) Natural Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• UXXX-AFM01 : MSci (hons) Natural Sciences (Full-time) - Year 3
• UXXX-AKM02 : MSci (hons) Natural Sciences with Professional Placement (Full-time with Thick Sandwich Placement) - Year 4
• UXXX-AAM02 : MSci (hons) Natural Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
Department of Physics
• USPH-AFB01 : BSc (hons) Physics (Full-time) - Year 3
• USPH-AFB05 : BSc (hons) Physics with Computing (Full-time) - Year 3
• USPH-AKB06 : BSc (hons) Physics with Computing (with Placement) (Full-time with Thick Sandwich Placement) - Year 4
• USPH-AAB06 : BSc (hons) Physics with Computing with Year Abroad (Full-time with Study Year Abroad) - Year 4
• USPH-AKB02 : BSc (hons) Physics (with Placement) (Full-time with Thick Sandwich Placement) - Year 4
• USPH-AAB02 : BSc (hons) Physics with Year Abroad (Full-time with Study Year Abroad) - Year 4
• USPH-AFB09 : BSc Physics (Full-time) - Year 3
• USPH-AKB09 : BSc Physics (with Placement) (Full-time with Thick Sandwich Placement) - Year 4
• USPH-AFM02 : MPhys Physics (Full-time) - Year 3
• USPH-AFM04 : MPhys Physics with Research Placement (Full-time) - Year 3

Notes:
* This unit catalogue is applicable for the 2012/13 academic year only. Students continuing their studies into 2013/14 and beyond should not assume that this unit will be available in future years in the format displayed here for 2012/13.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.