Page 103 - Calculus with Complex Numbers

But the Cauchy-lkiemann equations

bu gr bu gr
91 by gy 91
hold everywhere inside and on y . Hence we have

J flz) dz = J .fU) dz + J .fU) dz -F J .fU) dz -F J .fU) VZ = 0.
J / J /1 J /2 J /3 J /4

Case 2: y = .4rly rectangle Similar.
Case 3: y = .4rly rectilinear contour Meaning y = )(2XI yn where each yn is
a straight line parallel either to the real axis or to the imaginal'y axis.
N/ ?
By adding and subtracting ftlrther straight lines we canwrite y = X)1 yn where
y' are all rectangles. Therefore,

Case 4: y = .4rly closed contour We can choose a sequence of rectilinear
contotlrs yn, all lying inside y , such that