1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 339

y

Ci t t + Llt 13

0 x

Fig. 6.4

Definition 6. 7 The directional derivative off at z in the direction of the

arc is defined by

f~(z) = A-+0 lim

z+AzE7•

f(z + 6.z) - f(z)

6.z

= lim b.w/b.t = limAt-+ob.w/b.t

AHO 6.z/b.t z'(t)

(6.9-1)

provided that the limit in the last numerator exists. Of course, z'(t) f 0

for every t E [a:, ,BJ since 'Y is assumed to be regular. The notation 6.w / 6.t
in (6.9-1) is to be understood as an abbreviation for the ratio

f(z(t + 6.t)) - f(z(t))

6.t

Letting z'(t) = lz'(t)lei^9 , we may write (6.9-1) in the alternative form

f

'( ) = -ie limAt-+O 6.w/6.t

"Y z e lz'(t)I

(6.9-2)

Because of the dependence of the directional derivative on () = Arg z' ( t)
we shall use the notation f 0 (z) in preference to f~(z). An alternative
notation for the directional derivative is 8ef(z) [78).

Example We have seen that the function w = f(z) = lzl^2 = zz does not

have a derivative in the ordinary sense at any point z 0 f 0. Consider now

the arc"(: z = z(t) = t + it^2 (0:::; t :::; 2) and let z 0 = t 0 +it~. Then we have

f(z(to)) = t~ + t~

f(z(to + 6.t)) =(to+ 6.t)^2 +(to+ 6.t)^4

lim

...;...!-'-( z....:...( t_o _+_6._t )....:...) _-....:...f....:...( z-'"( t_o ):....:...) 3

= 2to + 4t 0

At-+O 6.t

z'(to) = 1+2ito