Page 52 - Calculus with Complex Numbers
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Ffgure 4. 7
Corollal'y 2 lf the closed contours n , yz are such that yz lies irlside n , and if
f (z) is differentiable on n , yz and between them, then
'
Proof lf we make cross cuts p, y4 as indicated in Figure 4.7 and if we denote
the upper parts of n , yz by y1'
'
)
:
, yz' and the lower parts by y) yM thenby Corollal'y 1
we have
Therefore
Corollal'y 3 lf non-intersecting closed contotlrs n , . . . , yn all lie inside the
closed contour y , and if f (z) is differentiable on y, n , . . . , yn and on the area
internal to y and external to n , . . . , yn, then (Figtlre 4.8)
Proof Make cross cuts as in the proof of Corollal'y 2.
