Begin2.DVI

Contour integration

Let C denote a curve in the z-plane connecting two points z=aand z=bas

illustrated in figure 12-13.

Figure 12-13. Curve in the z-plane.

The curve Cis assumed to be a smooth curve represented by a set of parametric

equations

x=x(t), y =y(t), ta≤t≤tb

The equation z=z(t) = x(t) + iy (t),for ta≤t≤tbrepresents points on the curve C

with the end points given by

z(ta) = x(ta) + iy (ta) = a and z(tb) = x(tb) + iy(tb) = b.

Divide the interval (ta, tb)into nparts by defining a step size h=

tb−ta

n and letting

t 0 =ta, t 1 =ta+h, t 2 =ta+ 2h,... , tn =ta+nh =ta+n

(tb−ta)

n =tb. Each of the

values ti, i = 0, 1 , 2 ,.. ., n, gives a point zi=z(ti)on the curve C. For f(z),a continuous

function at all points zon the curve C, let ∆zi=zi+1 −ziand form the sum

Sn=

n∑− 1

i=0

f(ξi)∆ zi=