Complex Analysis: An Introduction

Dr Kevin Houston follows up his best-selling book *How to Think Like a Mathematician* with *Complex Analysis: An Introduction*. Complex Analysis is a central subject in mathematics with applications in engineering, physics, and even the study of prime numbers. It has been said that often the shortest route in the solution of a real problem is to take a shortcut through the complex numbers.

Unlike other texts this book gets quickly to the heart of Complex Analysis: the concept of complex contour integration. This means that students get much more practice in the fundamental concept than they normally would. The central method of proof — use of the Estimation Lemma — is emphasised throughout because students then have a unifying principle to help understand and remember those proofs.

The book contains all you will need for an introductory course in Complex Analysis and includes a short and sweet proof of Cauchy’s Theorem — one which the majority of students can grasp not only the outline but the details as well.

The book contains copious examples and exercises tested on students arising from Dr. Houston’s 20 years plus experience of teaching the subject.

About the author

Kevin Houston

#### Why Complex Analysis?

Dr Houston says “I first met complex analysis when I went to university. It was an intriguing blend of calculus and complex numbers. It was a bit weird to begin with – you have a strange new definition of integration – but then the results began to be a lot simpler. For example, if a function is differentiable once, then it is differentiable an infinite number of times. That just doesn’t happen in calculus with real numbers. And it means that a Taylor series always exists. Not only that you could find really complicated real integrals with just simple ‘five-fingered’ exercises.”

“When I got my present job one of the first classes I was eager to teach was complex analysis. Since then, I’ve taught it to hundreds of students. So when deciding what to follow *How to Think Like a Mathematician* with, it was obvious. It had to be Complex Analysis!”

#### About the Author

Dr Houston has over 25 years of teaching experience, having taught at the University of Warwick, University of Liverpool, Middlesex University and is currently a Senior Lecturer at the University of Leeds.

Sample

Chapters

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Why you’ll

love it

#### Straight in

The central concept of contour integration is introduced as early as possible giving more time to practice and absorb it.

#### An easy Cauchy's Theorem

Cauchy’s Theorem produces all the great results but it’s proof is long and difficult in most books. Here we give a short and easy to understand proof.

#### Coherent

Almost all good theorems in complex analysis involve the Estimation Lemma. This is emphasised throughout.

#### Exercises

The best way to learn mathematics is to do it. Plenty of exercises, ranging from the easy to the challenging, are included in each chapter.

“The lectures were always really interesting and easy to follow”

**Third year student**

“Thanks for being an awesome lecturer and providing us with a lot of really useful resources. I wish most of my other lecturers would give us half what of what you did.”

**Kyle**

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