Complex Analysis - Maths KU

APP-4 Appendix II Proof of the Cauchy-Goursat Theorem
Note
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7.2
Appendix II Proof of the Cauchy-Goursat Theorem
In Section 5.3 we proved Cauchy’s theorem using Green’s theorem. This simple
proof was possible because of the hypothesis of continuity of f ′ throughout
a simply connected domain D. The French mathematician Edouard Goursat
(1858–1936) published a proof of Cauchy’s theorem in 1900 without this continuity
assumption. As a result his name was thereafter linked with Cauchy’s
in the title of one of the most fundamental of all theorems in complex analysis.
In this appendix we discuss how the full proof of the Cauchy-Goursat
theorem is accomplished. To avoid needless repetition throughout the following
discussion we will take for granted that we are working in a simply
connected domain D and that f represents a complex function analytic in D.
The proof of the Cauchy-Goursat theorem is accomplished in three steps.
The first two steps are helping theorems—sometimes called lemmas—which
are actually special cases of the Cauchy-Goursat theorem. The first of these
helping theorems deals with integrals along a triangular contour and the second
deals with integrals along a closed polygonal contour. The first theorem is
used in the proof of the second, and the second theorem is utilized to establish
the Cauchy-Goursat theorem in its full generality.
We will prove the first theorem, but because of its length, we will simply
sketch the proofs of the remaining two.
Theorem A.2 Triangular Contour
If ∆ is a triangular contour lying entirely within D, then �
∆
f(z) dz =0.
Proof Let ∆ be the triangular contour shown in Figure AII.1; the vertices
of ∆ are labeled A, B, and C. We form four smaller triangles C1, C2, C3,
and C4 by joining the midpoints E, F , and G of the sides of ∆ by straight line
∆
B C
Figure AII.1 Triangular contour
∆ within D
A
C 1
∆
C 4
B F
C
A
C 3
E G
C 2
Figure AII.2 Triangular contours
C1, C2, C3, and C4