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Warning: This opinion is written for people who are doing degree courses, or equivalent, in maths or physics and are about to take courses in complex analysis. The subject of 'Complex Analysis' usually forms part of the core for maths degrees and often turns up in mathematical physics courses as well so if you are studying either of these, then the chances are that you will come across it sooner or later. I am working towards a Phd in physics and have found that this subject is of incredible use in what I will be doing in the future. So what is complex analysis you may ask? Well to start off with it doesn't mean hard or difficult analysis, instead what it deals with are functions of complex variables (variables that can be complex numbers) that have a derivative (or are analytic in the language of the field). When you start the subject it seems as if everything is similar to the standard real analysis or basic calculus courses that you may have done in the first year - but in fact things are totally different. For example - if a function of a complex variable has a first derivative then it has all the higher derivatives as well - something that is not the case in standard calculus. Ok then - on to the book itself. First of all it is written with Physicists and Engineers in mind which in my opinion makes it a lot easier to read then texts written for mathematicians. Saying that though I think mathematicians would also find it very suitable as it is quite rigorous and offers full proofs of all of the theorems presented (it is also on the reading list of many university maths departments). By the time the average student is faced with these courses they will have already covered the basics of complex numbers but if they are a little rusty then the first chapter serves as a nice revision aid. There are lots of examples and exercises to make sure that you are speaking the language of the rest of the book before you s
tart the rest of it. One of the nice things about the book is that it often reminds you of things that you should have learned from previous courses before going on to use that knowledge in the context of complex analysis. Personally I found this a great help and I imagine that most students would too because it could have been over a year since some of the older stuff was last seen. The text includes all the usual topics of introductory complex analysis courses including The complex function and it’s derivative, the Cauchy-Riemann equations, the complex versions of transcendental functions such as sine and cosine, Integration in the complex plane (including Cauchy’s theorm and Cauchys Integral formula),fractals, Infinte Series (including Taylor series and Laurent series), Residues and their use in real integrals, Laplace Transforms and Conformal Mapping. As well as covering all of the theory this book includes many applications of the subjects in applied sciences such as physics and engineering – with a heavy leaning to electrical engineering since that is the original subject of the author. I found these very useful and interesting, partly as practice of the theory and partly as subjects in their own right. I particularly liked the short section on fractals. There are also quite a large number of worked examples – which is great help for battling through homework and exam problems – as well as a good number of exercises. In short I believe that this book is probably the best I have ever seen on the subject and I would recommend it to anyone who is about to take courses in complex analysis or who needs a refesher course in the subject. The only down point is the high price (the one I have given is from Amazon at the time of writing) - but it is worth every penny in my belief.
Complex Variables with Applications remains accessible to students of engineering and mathematics with varying mathematical backgrounds. Designed for the one-semester course in complex analysis, there is optional review for students who have only studied calculus and differential equations. New features of this edition include: wider breadth of topic coverage; improved exercise and problem sets; strong, up-to- date engineering applications; and more thorough coverage of applications and motivation of theory.