Page 8 - Calculus with Complex Numbers

Preface

This book is based on the premise that the learning curve is isomorphic to the
historical curve. ln other words the learning order of events is the same as the
historical order of events. For example we learn aritluueticbefore we learn algebra.
W e learn how before we learn why.
Historically, calculus with real numbers came lirstm initiated by Newton and
Leibnitz in the seventeenth centuly Hven though complex numbers had been
known about from the time of Fibonacci in the thirteenth centtlry, nobody thought
of doing calculus with complex numbers until the nineteenth centuly Here the
pioneers were Cauchy and Riemann. Rigorous mathematics as we know it today
did not come into existence until the twentieth centuly lt is important to observe
that the nineteenth centul'y mathematicians had the right theorems even if they
didn't always have the right proofs.
The learning process proceeds similarly. Real calculus comes lirst followed by
complex calculus. ln lnoth cases we learn by using calculus to solve problems. lt
is when we have seen what a piece of mathematics can do that we begin to ask
whether it is rigorous. Practice always comes before theoly
The emphasis of this book therefore is on the applications of complex calculus
rather th= onthe foundations of the subject. A working knowledge of real calculus
is assumed also an acquaintance with complex numbers. A background not unlike
that of an average mathematician in 1800. Hquivalently, a British student just
starting at university. The approach is to ask what happens if we try to do calculus
withcomplex numbers instead of withreal numbers. W e lindthatparts arethe same
whilst other parts are strikingly different. The most powerful result is the residue
theorem for evaluating complex integrals. Students wishing to study the subject at
a deeper level should not lind that they have to unlearn anything presented here.
1 would like to tha111: the mathematics students at M anchester University for
sitting patiently through lectures onthis material over the years. Also for their feed-
back (positive and negative) which has been invaluable. The book is respectfully
dedicated to them.
Jolm (B . Reade
June 2002