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
cos^{2}
q = 1 – sin^{2}
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
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
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