1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

· [x(tk) - x(tk-1) + i(y(tk) - y(tk-1))]

= if3 [u(x(t), y(t)) dx(t) - v(x(t), y(t)) dy(t)]

• i 1: [v(x(t), y(t)) dx(t) + u(x(t), y( t)) dy(t)]

= j ( u dx -v dy) + i i ( v dx + u d~)

'Y

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In the case of a continuously differentiable arc, or of a piecewise con-

tinuously differentiable arc, formulas (7.8-19) and (7.8-20) are applied to

evaluate the corresponding line integrals. Formula (7.8-14) may be used

to evaluate a complex integral in terms of the two components real line

integrals. Also, it may be used to derive a number of properties of the

complex integral.

Example To find J'Y z^2 dz,. where "(: z = t - it^2 , 0 S t S 1. In this case

u = x^2 -y^2 , v = 2xy, x = t, y = -t^2 • Hence·

j z^2 dz= j[(x^2 - y^2 ) dx - 2xy dy] +if [(2xy dx + (x^2 - y^2 ) dy]

'Y 'Y 'Y

= fo\t2 -5t^4 ) dt + i fo-4t^3 + 2t^5 ) dt

= - ~ (1 + i)

3

7.9 Cauchy's Fundamental Theorem

The following theorem, due to A. L. Cauchy [7], is of fundamental
importance in the theory of complex integration.

Theorem 7.11 Suppose that f is analytic in a simply or ,multiply con-

nected region R, that f' is continuous in R, and let 0 be a simple closed

contour homotopic to a point in R. Then

j f(z)dz = 0 (7.9-1)

c

where 0 is described once in either direction (Fig. 7.5).

Remark In view of Theorem 6.37 it is clear that the hypothesis of the
continuity of f' in R is superfluous. The following proof of Cauchy's the-

orem requires the continuity of f'. E. Goursat [16] in 1900 showed that