The Cauchy Theory: A Fundamental Theorem
As with the theory of differentiation for complex-valued functions of a complex variable, the integration theory of such functions begins by mimicking and extending results from the theory for real-valued functions of a real variable, but again the result
- PDF / 462,284 Bytes
- 37 Pages / 439.36 x 666.15 pts Page_size
- 45 Downloads / 177 Views
The Cauchy Theory: A Fundamental Theorem
As with the theory of differentiation for complex-valued functions of a complex variable, the integration theory of such functions begins by mimicking and extending results from the theory for real-valued functions of a real variable, but again the resulting theory is substantially different, more robust, and more elegant. Specifically, a curve or path in C is a continuous function from a closed interval in R to C. Thus the restriction of a complex-valued function f on C to the range of a curve has real and imaginary parts which can be viewed as real-valued functions of a real variable and thus integrated on the interval.1 Adding the integral of the real part to { times the integral of the imaginary a complex-valued integral R R part defines R of a complex-valued function (i.e., f D 0, with g 2 C1 .R/. Let n 2 Z>0 and define ./ D g./e{ , where 2 Œ0; 2 n. Assume that g.0/ D g.2 n/. Observe that the conditions on g imply that the curve winds around the origin n times in the counterclockwise direction and, as expected, I.; 0/ D
1 2{
Z Z
1 dz D z 2{
Z
2 n 0
d.g./e{ / g./e{
g 0 ./e{ C {g./e{ d g./e{ 0 Z 2 n 0 g ./ 1 C { d D n: D 2{ 0 g./
D
1 2{
2 n
In general, let W Œa; b ! C fcg be a continuous closed path and let f be a dz primitive of on . Then f .t/ agrees with a branch of log..t/ c/; that is, zc ef .t / D .t/ c for all t 2 Œa; b: Hence I.; c/ D
f .b/ f .a/ : 2{
We see that the C1 assumption on g is unnecessary as we will also be able to conclude using homotopy of curves discussed in the next section.
4.4 Homotopy and Simple Connectivity
97
4.4 Homotopy and Simple Connectivity In order to give the integration results the clearest formulation (see for instance Corollary 4.52), we introduce the topological concepts of homotopic curves and simply connected domains. Definition 4.40. Let 0 and 1 be two continuous paths in a domain D, parameterized by I D Œ0; 1 with the same end points; that is, 0 .0/ D 1 .0/ and 0 .1/ D 1 .1/. We say that 0 and 1 are homotopic on D (with fixed end points) if there exists a continuous function ı W I I ! D such that (1) (2) (3) (4)
ı.t; 0/ D 0 .t/ for all t 2 I ı.t; 1/ D 1 .t/ for all t 2 I ı.0; u/ D 0 .0/ D 1 .0/ for all u 2 I ı.1; u/ D 0 .1/ D 1 .1/ for all u 2 I
We call ı a homotopy with fixed end points between 0 and 1 ; see Fig. 4.3, with u .t/ D ı.t; u/, for fixed u in I . Let 0 and 1 be two continuous closed paths in a domain D parameterized by I D Œ0; 1; that is, 0 .0/ D 0 .1/ and 1 .0/ D 1 .1/. We say that 0 and 1 are homotopic as closed paths on D if there exists a continuous function ı W I I ! D such that (1) ı.t; 0/ D 0 .t/ for all t 2 I (2) ı.t; 1/ D 1 .t/ for all t 2 I (3) ı.0; u/ D ı.1; u/ for all u 2 I The map ı is called a homotopy of closed curves or paths; see Fig. 4.4, with u .t/ D ı.t; u/ for fixed u in I . A continuous closed path is homotopic to a point if it is homotopic to a constant path (as a closed path). γ0(t)
Data Loading...