Introduction to Complex Analysis

Lecture notes for Advanced Mathematics for Physics, PHYS20041

Preface

These pages form the lecture notes for the Complex Analysis component of Advanced Mathematics for Physics (PHYS20041).

Errata and misunderstandings

If you discover any mistakes in these notes, no matter how small, please inform me via email or via the unit discussion boards. If there is a spelling mistake or some bad punctuation, I would like to know and to correct it. If there is a mathematical error, then definitely let me know — as a teacher, the last thing I want to do is confuse my students with bad mathematics.

If you find a part of the notes that you don’t understand, again feel free to contact me. I may be able to provide an alternative explanation that works better for you. Alternatively, perhaps that part of the notes could be clarified, or perhaps you have found an error in the notes without realising it. In any case, by requesting help to understand something in the notes, you may also be helping me to improve them.

I will endeavour to fix the lecture notes as any errors are found. Any recent edits will be listed below, so that you are alerted to any changes.

Recent fixes and clarifications

• (12/01/2025) Fixed the definition of addition in ‘Chapter 0’. Naturally, \((a + bi) + (c + di)\) is equal to \((a + c) + (b + d)i\), not \((a + b) + (c + d)i\).
• (13/01/2025) Added a continuation to the definition of \(z^n\) in section 2.3 for the case where \(n\) is complex.
• (22/01/2025) Added an extra step in the description of theorem 4.1 (differentiation of a power series).
• (16/02/2025) Corrected example 7.4 so that region \(B\) is described as \(\left|z\right| > 1\) not \(\left|z\right| > 2\).
• (16/02/2025) Corrected the last line of equation 7.4 so that the sum goes from \(n = -\infty\), not \(n = \infty\).
• (04/04/2025) Corrected the definition of the first branch of \(f(z) = (z - 1)^{1/3}\) in example 5.2, so that it isn’t identical to the third branch. Attempted to clarify the following explanation.

Notation

I use \(z^*\) to represent the complex conjugate of \(z\), rather than \(\overline{z}\). (Either is fine in your work.) I use \(\log\) to mean the natural logarithm¹. If I use subset notation, I will use \(A \subseteq B\) to mean that \(A\) is a subset of \(B\), with the possibility that \(A = B\), while \(A \subset B\) will mean that \(A\) is a strict subset of \(B\). Meanwhile, \(A \setminus B\) means the set difference, e.g. \(\mathbb{C} \setminus \{0\}\) is the set of all complex numbers except zero.

¹ Historically, logarithms to base 10 were used a lot as a calculational aid, but today, with the advent of the pocket calculator and the computer, these are encountered by physicists a lot less frequently that the natural logarithm. Interestingly, Python also uses \(\log\) to mean the natural logarithm.

Acknowledgements

These notes are certainly influenced by those of my predecessor, Dr. Alex Jones, who taught Complex Analysis in the Methods of Theoretical Physics unit in 2021-22. I should also acknowledge H. A. Priestley, author of “Introduction to Complex Analysis” from which I learnt the subject many years ago and to which I still refer when needed. For those of you who wish to improve their understanding of the mathematical foundations of the subject or who simply wish to study further, I do not hesitate to recommend this book.