1550251515-Classical_Complex_Analysis__Gonzalez_

136 Chapter^3

or

(6) By properties 2 and 5 we have

lim [f(z) + J(z)] = L + L,

z-+a

lim[2Ref(z)] = 2ReL,

z-+a

which gives

lim Ref(z) = ReL,

z-+a

lim[f(z) - f(z)] = L - L

z-+a

lim[2ilmf(z)] = 2ilmL

z-+a

lim Imf(z) = ImL (3.7-1)

z-+a

If a is finite and we write z = x + iy,a = a 1 + ia 2 ,L = L1 + iL 2 ,f(z) =
u ( x, y) + iv ( x, y), the preceding result can be expressed as follows:

lim v(x,y) = Lz

( x ,y )-+( ai ,a.)

(7) If (3.7-1) holds, using properties (1), (2), and (3), we obtain

lim z-+a J(z) = z-+a lim Ref(z) + i z-+a lim Imf(z) = L 1 + iL 2 = L

(8) This property follows at once from the inequality

llJ(z)l - ILll ~ IJ(z)-LI

(9) For any given E > 0, if E ;::^1 krr, choose E^1 such that 0 < E^1 <^1 krr,

and if E < %'Tr, let E^1 = E. Next, choose 0 < f3 ~ ILi sin E^1 and such that

Np(L) be contained in the principal region z + lzl f 0, i.e., such that

0 < f3 < I Im LI (Fig. 3.8). By definition of limit, corresponding to this f3

there is a 8 > 0 such that

f(z) E Np(L) whenever z E N~(a) n D

Then if a is the smallest (positive) angle formed by the direction of the

vector L with either tangent line from 0 to the circle I( - LI = /3, it follows

w-plane

Arg L

u

Fig. 3.8