Lemma: Sum(k=0,k=n-1) z^k.w^(n-k-1) = (z^n - w^n)/(z - w) for z <> w Proof. Let: f_n(z,w) = Sum(k=0,k=n-1) z^k.w^(n-k-1) z.f_n(z,w) = Sum(k=0,k=n-1) z^(k+1).w^(n-k-1) = z.w^(n-1) + z^2.w^(n-2) + .. + z^(n-1).w + z^n w.f_n(z,w) = Sum(k=0,k=n-1) z^k.w^(n-k) = w^n + z.w^(n-1) + z^2.w^(n-2) + .. + z^(n-1).w ---------------------------------------------------------------------- (z - w).f_n(z,w) = - w^n + z^n Lemma: Sum(k=0,k=n-1) z^k.z^(n-k-1) = f_n(z,z) = n.z^(n-1) for z = w Define: F(z,w) = Sum(n=0,n=oo) c_n . f_n(z,w)Published in 'sci.math':
Lemma 2.1. (ML Inequality) on page 18: L is the length of γ . But this can only be the case if γ is not, for example, a Space-filling curve, with "infinite" length L . Curves called rectifiable are mentioned, but not explained, in the book on page 16.
Example on page 20 shows that ∫γ
dz / z = 2π.i where γ : [0,2π] => C is γ(t) =
r.ei.t with real r > 0 .
Nice place to hook in and show that, with the same closed curve (being a circle with radius r) :
∫γ zm dz = 0 or 2π.i and the latter only for m = -1 , where m is an integer. Enhance and repeat:
So, indeed it is in some sense hust about the only way to get something nonzero by integrating an analytic function with an "isolated singularity" over a closed curve.
On the same page, it reads: we have not given a definition of what it means for a point to lie "inside" a closed curve. In fact we will not be giving any such definition. Everybody seems to be affraid of the Jordan curve theorem. As a consequence of the paranoia, concepts which are intuitively clear - like "inside / outside" - are rather dismissed as "undefined". Irritating in the first place. And not quite neccessary as well. It makes content of CMS more difficult to absorb. And it distracts attention from issues more relevant than this.
Worse. Spending so much effort on the generality of (closed) curves needed to evaluate the complex line integrals, it sounds like a worthwile undertaking, but it isn't. The reason is the following. A theory called Lissajous Analysis shows that any closed smooth curve can be represented by a Fourier Series, which is the same as a Complex Laurent Series specified for the unit circle. The reverse is also true. Any complex Laurent series, when specified for the unit circle, represents a closed smooth curve. Actually it means that a line integral over an arbitrary smooth closed curve can always be converted into a line integral over the unit circle:
∫γ f(z) dz = ∫O f(γ(z)) γ'(z) dz
An argument remotely similar to the above is found in CMS on page 80 as Theorem 5.1 (The Argument Principle). Oh well, not even remotely, it seems ..
So what are we interested in? More complicated contours or complicated complex
functions? I wouldn't vote for the extreme and ONLY admit integrations over the
unit circle. But the above shows that a constraint to, say, simple closed curves
is very much reasonable. And that the reverse strategy - complicated contours -
impressive as it may seem, is in fact rather futile: a parametrization issue.
Mathematics can be too general for comfort. If the generalization of a special result is more or less trivial, then show some restraint and go for the specialization.
Example. Let γ(z) = zn with n ∈ N , then γ is prototype of a curve winding around itself (winding number =) n times and:
∫γ dz / z = ∫O n zn-1 / zn dz = n ∫O dz / z = n 2 π i
Theorem 2.2 (Cauchy-Goursat: Cauchy's Theorem for triangles). Ullrich's
Proof on page 23 starts with "Suppose not." : thereby suggesting a proof
by contradiction. But only suggesting. The proof is actually a straightforward
one, i.e. a direct proof.
It is noted that the (quite ingenious) proof of Cauchy-Goursat uses the fact that the absolute value of the complex derivative is maximal somewhere inside the triangles. Such is for example not the case with the function f(z) = 1/z . Thus the theorem need not to be valid for such a function (actually it isn't). The reverse is not true. If a function is singular, then the integral around the singularity can (easily) be zero.
Example. f(z) = 1/z2 => ∫|z|=1 dz / z2 = 0 .
On page 23 it also reads that you can get the conclusion from Green's theorem , i.e. for complex (thus real) differentiable functions within simply closed (Jordan) curves:
Note. The Least Squares Finite Element Method for Ideal Flow can be interpreted as the Cauchy Integral Theorem applied to Numerical Analysis. Take a look at page 6 of the article when seeking for confirmation of this fact (Green's Theorem being the would-be real valued equivalent of Cauchy's).
Published in 'sci.math':
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while investigating the initial polynomial 1 + z + z2 / 2! + z3 / 3! + z4 / 4! ... + zn / n! of the entier function ez ,
with help of Rouché's Theorem (I love this Wikipedia source. Because now it has become my theorem as well)
Corollary 3.7 (Fundamental Theorem of Algebra) on page 37. Quote:
for us it's just an amusing digression, at least for now. Back to work.
The following formula, expressing any complex polynomial in its roots, is even
between scaring parentheses:
Why oh why, this seemingly disdainful treatment of a most impressive result? Has the author forgotten that this is the main thing Complex Analysis has been invented for in the old days? A preliminary version of The Argument Principle can be derived at this place. It's almost as if we have jumped into
Chapter 5. Counting Zeroes and the Open Mapping Theorem , with:
Theorem (remotely resemblant to the Picard theorems).
A polynomial of degree n assumes all complex values c exactly n times.
Proof. an zn + an-1 zn-1 + ... + a2 z2 + a1 z + a0 - c = 0 has n zeroes.
A reasoning as in the last paragraph on page 40 of CMS could also be employed for other "Special Functions" than f(z) = exp(c.z) : f(z) = c.z ; f(z) = c.ln(z) ; f(z) = zc .
Open mapping theorem (complex analysis)