PHYS 20672 Complex Variables and Vector Spaces

• Textbooks: I recommend Spiegel et al Complex Variables (in the Schaum's outlines series), Arfken & Weber Mathematical Methods for Physicists or Riley, Hobson & Bence Mathematical Methods for Physics and Engineering. For much of the material on vector spaces of finite dimension, Riley is a useful reference. The first chapter of R. Shankar, Principles of Quantum Mechanics, may be more generally useful; it can be found on Blackboard.

• The last examples class was at 10.00 on Monday 13 May in the Braddick Library rooms L1-L3, Floor 1 Schuster Building. In the run-up to the exams, it should usually be possible to find me in my office (7.17 Schuster Building).

• Lecture summaries (for lectures 1-16, refs to Riley refer to the first edition, not the third edition available online -- will be corrected in time):
  • Lectures 1-4
  • Lectures 5-7
  • Lectures 8-12
  • Lectures 13-16
  • Lectures 17-21 (from this point onwards, "Riley" refers to the third edition, available online)
  • Lectures 22-23

• Scans of the visualizer slides are available on Blackboard.

• Downloads:
• Mappings of the complex plane defined by some simple functions.
• An illustration of the argument theorem.
• Solutions of Laplace's equation obtained by conformal mapping.

• An online tool to help with visualizing conformal mappings. From the same source you might also like world map projections. Note that when you have one shape-preserving mapping from the sphere to the x,y plane (e.g., Mercator's projection), you can obtain many others by conformal mapping. So if you like maps, this could be an important application of functions of a complex variable!

• Examples sheets and worked solutions:
• Examples 1 and Solutions 1 (released 2019)
• Examples 2 and Solutions 2 (released 2019)
• Examples 3 and Solutions 3 (released 2019)
• Examples 4 and Solutions 4 (released 2019)
• Examples 5 and Solutions 5 (released 2019, solutions to Q.42 clarified 13.05.2019)
• Examples 6, Solutions 6, Part 1 and Solutions 6, Part 2 (released 2019)
• Examples 7 and Solutions 7 (released 2019)
My view on maths examples is that it is better to get more practice rather than less. However, some questions intended partly to extend your understanding have been marked with one or two dieses (‡ or ‡‡). In the early sheets these are not usually harder than other problems, but you may want to give them lower priority. Problems marked with an asterisk (*) are not intended to be lower priority, but they may be more challenging; I recommend that you try them, but set yourself a time limit.

• The format of the exam is similar to previous years' papers for PHYS20672: one compulsory question (consisting of several short questions covering the whole course) and a choice of two out of three longer questions. Two of the longer questions will be on complex variables and one will be on vector spaces.

• For practice on short past exam questions on vector spaces, see
  • PHYS30201: 2016/17 1(a); 2015/16 1(b); 2014/15 1(a),(b),(d); 2013/14 1(a) [1(e) should also be OK, with the understanding that x is position and that momentum p is related to k by a factor of ℏ).
  • PHYS20602: 2011/12 1(a),(d)(i),(d)(ii); 2010/11 1(a),(b),(d); 2009/10 1(a); 2008/09 1(a). [Local copies are provided in the two cases where it's painful to extract the papers from Blackboard -- the problem may have been fixed by now.]
  For practice on longer questions (and for a wider variety of short questions) you should study examples sheets 6 and 7 (but not questions or parts of questions marked with a diesis, ‡).

• Please let me know if you find errors in the summaries and examples sheets/solutions.