It’s coming…...

Posts have been a bit scarce round here recently. Something’s coming soon…

Common Mistakes in Complex Analysis (Revision help)...

My new book, Complex Analysis: An Introduction, is nearly finished. To help my students with revision I created a list of common mistakes and this forms a chapter in the book. As a lecturer with many years of experience of teaching the subject I have seen these mistakes appear again and again in examinations. I’m sure that, due to pressure, we’ve all written nonsense in an exam which under normal conditions we wouldn’t have. Nonetheless, many of these errors occur every year and I suspect something deeper is going on. What follows is not intended to be a criticism of my students, who, luckily for me, are generally hard-working and intelligent. Nor is it an attempt to mock or ridicule them. Instead the aim is to identify common mistakes so that they are not made in the future. And if this post seems negative in tone, the a later one is more positive as it delves into techniques that improve understanding. Imaginary numbers cannot be compared The first mistake is the probably the most common: the comparison of imaginary numbers. For example, students write for a complex number. This cannot be right. If were what does it mean for to be less than ? What is usually intended is the modulus of , i.e., . The point is, unlike real numbers, we cannot order the complex numbers. For example, which is bigger or ? This is difficult to decide! Since complex numbers can be identified with the plane ordering them is equivalent to ordering the points of the plane and clearly this can’t be done — at least not in any useful or meaningful way. One last point needs to be made. Although is incorrect, note that expressions like can be true if is...

Best of xkcd: Educational...

The web comic xkcd is well-known for its humour. I’ve also got a soft spot for the educational infographics, such as the following. (Not all the information should be...

Sequences and Series

I’ve been working on other things for a while but have started to get moving on my latest project: Sequences and Series. This began with the leftovers from How to Think Like a Mathematician. I had written quite a few chapters on analysis, in particular, on the definition of the limit of a sequence. After all, the concept of epsilon-delta proof is a key topic to grasp in mathematics and require deep logical thinking for most students. Anyhow, the chapters had to go as the book would have been too long. I had intended to use them as a basis for a book on real analysis but got involved in a book on complex analysis. Next, one of my colleagues produced a book on real analysis! So rather than be ungentlemanly direct competition for him I’ve used my notes to create a website on Sequences and Series. It’s a bit basic at the moment (and has been up for a while) but I’ll add more this year. A book will follow in due...

Gresham/LMS Lecture

This week the joint Gresham College and London Mathematical Society lecture will take place. Reidun Twarock of the University of York will give a talk Geometry: A New Weapon in the Fight Against Viruses. The details are on the Gresham College website. In case you can’t make the talk on Wednesday, then maybe there is a MathsJam near you on Tuesday. And if you can’t make either, then maybe you would like to see last year’s Gresham/LMS talk by Marcus du...