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...

Nate Silver on Panorama

I’m not a regular watcher of Panorama. It is a TV programme that has dumbed down considerably in my lifetime and I have probably only seen two programmes in the past decade. One was a couple of weeks ago on male suicide and the other was this Tuesday’s episode about Nate Silver attempting to predict the outcome of the forthcoming General Election here in the UK. Nate Silver’s book The Signal and the Noise, on the uses of statistical methods for prediction, is one of my favourite books of recent years and is well worth reading if you haven’t read it. This programme is not so good. Silver was successful in predicting, against a backdrop of pundits who said it was too close to call, that Obama would win the last US Presidential election. Hence, it is natural to see if he can do the same in a UK General Election. My recollection is that he discusses this in his book and points how it is much much harder it is. Given recent developments in the UK it is now even harder… Anyhow, the programme itself is rather lightweight. The presenter Richard Bacon takes Silver round the country explaining the UK system, Silver goes back to US, and makes a prediction. There is no explanation of the statistical methods involved or how they relate to the election. How Silver arrives at the conclusion is overlooked. I’m not expecting an explanation of Bayesian inference or some such on prime time TV but the probabilities and ranges in his prediction have been removed and replaced with absolute numbers. An averagely intelligent viewer can grasp the concept of range surely. Hence, no major insight is gained. The prediction is not that different to what many would guess. My prediction was that, due to soft UKIP support, the Conservatives would be largest party but no majority, and that Labour and SNP would have a combined majority. And, apart from the numbers in the centre of the ranges, that’s all that the programme gives us. That’s a bit disappointing. The programme is available for 11 months on iPlayer but I guess that it is not available in all territories. Watch the programme here. Silver’s report, made in conjunction with three British academics, is available on Silver’s site The Five Thirty Eight. The three academics, Chris Hanretty, Benjamin Lauderdale, and Nick Vivyan, maintain the site electionforecast.co.uk. On the subject of election results, why do newspapers never give us the number of Don’t Knows when they publish a poll result before the election? I would say that group is important. Also, the number of Don’t Knows can be as large as 40% at the start of a campaign. It would be interesting to see how this changes during an election and to know why they are Don’t Knows. Just had to get that grumble off my chest… UPDATE: Well, just about everyone got it wrong. The Conservatives achieved a majority government! David Spiegelhalter has quite a bit to say about the polling problems on his Understanding Uncertainty...