[U_Complex20151117] Lagrange Identity for Complex Numbers

Question

Introduction

     I was revising my complex analysis just for fun (I had graduated almost 3 decades ago) and I came across this problem which is question 5 from page 9 of the book Complex Analysis by Ahlfors.  This book is pretty hard core for a first course in complex analysis, but this is not surprising since Lars Ahlfors was no less than a Field’s medallist.

     Intuitively, I knew that this identity looks like the vector identity

                                        |a · b|² = |a|² |b|² – |a ´ b|²

and if one strips away the |a|² |b|², it boils down to the Pythagorean Trigonometric Identity

                                         cos² q = 1 – sin² q

I thought if I could just define the appropriate dot and cross products (actually this can be done), I could solve it easily.  However, this itself requires proof.  I was begging the question.  In fact, it is precisely because of this Lagrange Identity (and the related Cauchy-Schwarz Inequality) that allows  cos q  and   sin q  to be meaningfully defined.

     OK, it looks like I have to do it the hard way.  The tricky part in the manipulation of those sums in sigma notation is to ensure, at each step, that I did not introduce any spurious terms, nor miss out any terms.  To simplify the notation, in what follows I shall assume that  i  and  j  are indices in the range of whole numbers  [n] = {1, ... , n}.

Solution

Remarks

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* University / College level Complex Analysis

* other syllabuses that involve Complex Analysis

* any independent learner game for a challenge