PlanetMath: proof of Goursat's theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

proof of Goursat's theorem

(Proof)

We argue by contradiction. Set

$\displaystyle \eta = \oint_{\partial R} f(z) dz,$

and suppose that $\eta\neq 0$ . Divide

into four congruent rectangles

(see Figure 1), and set

$\displaystyle \eta_i = \oint_{\partial R_i} f(z) dz.$

$\includegraphics[scale=.7]{ProofGoursatTheorem.eps}$

Figure 1: subdivision of the rectangle contour.

Now subdivide each of the four sub-rectangles, to get 16 congruent sub-sub-rectangles $R_{i_1i_2},\; i_1,i_2=1\ldots 4$ , and then continue ad infinitum to obtain a sequence of nested families of rectangles $R_{i_1\ldots i_k}$ , with $\eta_{i_1\ldots i_k}$ the values of integrated along the corresponding contour.

Orienting the boundary of and all the sub-rectangles in the usual counter-clockwise fashion we have

$\displaystyle \eta = \eta_1+\eta_2+\eta_3+\eta_4,$

and more generally

$\displaystyle \eta_{i_1\ldots i_k} = \eta_{i_1\ldots i_k1}+\eta_{i_1\ldots i_k2}+\eta_{i_1\ldots i_k3}+\eta_{i_1\ldots i_k4}.$

In as much as the integrals along oppositely oriented line segments cancel, the contributions from the interior segments cancel, and that is why the right-hand side reduces to the integrals along the segments at the boundary of the composite rectangle.

Let $j_1\in\{1,2,3,4\}$ be such that $\vert\eta_{j_1}\vert$ is the maximum of $\vert \eta_i \vert, i=1,\ldots,4$ . By the triangle inequality we have

$\displaystyle \vert\eta_1\vert+\vert\eta_2\vert+\vert\eta_3\vert+\vert\eta_4\vert \geq \vert\eta\vert,$

and hence

$\displaystyle \vert\eta_{j_1}\vert\geq 1/4 \vert\eta\vert.$

Continuing inductively, let $j_{k+1}$ be such that $\vert\eta_{j_1\ldots j_k j_{k+1}}\vert$ is the maximum of $\vert\eta_{j_1 \ldots j_k i}\vert, i=1,\ldots,4$ . We then have

$\displaystyle \vert\eta_{j_1\ldots j_k j_{k+1}}\vert \geq 4^{-(k+1)} \vert\eta\vert.$

(1)

Now the sequence of nested rectangles $R_{j_1 \ldots j_k}$ converges to some point $z_0\in R$ ; more formally

$\displaystyle \{z_0\} = \bigcap_{k=1}^\infty R_{j_1\ldots j_k}.$

The derivative

is assumed to exist, and hence for every $\epsilon>0$ there exists a

sufficiently large, so that for all $z\in R_{j_1\ldots j_k}$ we have

$\displaystyle \vert f(z)-f'(z_0)(z-z_0) \vert \leq \epsilon \vert z-z_0\vert.$

Now we make use of the following.

Lemma 1 Let $Q\subset\mathbb{C}$ be a rectangle, let $a,b\in\mathbb{C}$ , and let be a continuous, complex valued function defined and bounded in a domain containing . Then,

		$\displaystyle \oint_{\partial Q} (az+b) dz = 0$
		$\displaystyle \left\vert \oint_{\partial Q} \!\!\!f(z) \right \vert \leq MP,$

where is an upper bound for $\vert f(z)\vert$ and where is the length of $\partial Q$ .

The first of these assertions follows by the Fundamental Theorem of Calculus; after all the function

has an anti-derivative. The second assertion follows from the fact that the absolute value of an integral is smaller than the integral of the absolute value of the integrand -- a standard result in integration theory.

Using the Lemma and the fact that the perimeter of a rectangle is greater than its diameter we infer that for every $\epsilon>0$ there exists a sufficiently large that

$\displaystyle \eta_{j_1\ldots j_k} = \left\vert \oint_{\partial R_{j_1\ldots j_... ...t\partial R_{j_1\ldots j_k}\vert^2 = 4^{-k} \vert \partial R\vert^2 \epsilon.$

where $\vert\partial R\vert$ denotes the length of perimeter of the rectangle

. This contradicts the earlier estimate (1). Therefore $\eta=0$ .

"proof of Goursat's theorem" is owned by rmilson.

(view preamble | get metadata)

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: estimate, diameter, perimeter, theory, absolute value, fundamental theorem of calculus, length, upper bound, domain, bounded, function, complex, continuous, derivative, point, converges, triangle inequality, composite, side, segments, interior, line segments, oriented, integrals, boundary, sequence, ad infinitum, contour, subdivision, rectangles, congruent, divide, contradiction

This is version 10 of proof of Goursat's theorem, born on 2002-08-02, modified 2003-07-15.
Object id is 3261, canonical name is ProofOfGoursatsTheorem.
Accessed 5265 times total.

Classification:

AMS MSC:

30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact

post | correct | update request | add example | add (any)