Page 101 - Calculus with Complex Numbers
P. 101
Take flz) = .:4 + 1, g (c) = c. The zeros of f (z) are at û?, a?, û?5, û)7
where û? = eiICI4 Let y = n + yz + p where yl is the straight line z = .x
,
(0 s .x s R), yz is the arc c = eit (0 s t :s(:v/2), and p is the straight line
z = iy (R k: y k: 0).
W e have .x4 + 1 >
. x . (Clearly! )
On p'z
Hence Iflz) I> Ig (z) Ion y if R > 2. Therefore f (z) = ./ + 1,
flz) + g (z) = .:4 + c + 1 have the same number of zeros inside y if R > 2,
namely 1. Argue similarly for the other quadrants.
