1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

6.10 EXPRESSION OF THE DIRECTIONAL

DERIVATIVE IN TERMS OF fz, f-z, AND 6

341

Suppose that w = f(z) = u +iv is of class 'D(A), A open, and let 1* CA,

where 1: z = z(t) = x(t) + iy(t), a::::; t::::; /3, is a regular arc. Let z E 1*

be a fixed point, and z + l::.z = z(t + !::.t), t + t::.t E [a,/3]. Then we have

l::.z(t) = l::.x(t) + it::.y(t), and if we write

z'(t) ~ x'(t) + iy^1 (t) = lz'(t)lei^8

it follows that

x'(t)

lz'(t)I =cosO,

y'(t).

lz'(t)I = smO

as we have seen in Section 3.13.

Let !::.w = !::.u + i l::.v be the increment in w corresponding to l::.z. From

l::.u = Ux l::.x + Uy l::.y + c1 l::.x + c2 l::.y

l::.v = Vx l::.x +Vy l::.y + c3 l::.x + c4 l::.y

we obtain
t::.w. t::.x. t::.y l::.x t::.y

!::.t =(ux+ivx) !::.t +(uy+ivy) !::.t +.A1 !::.t +.A2 !::.t

where .A 1 = c 1 + ic 3 and .A 2 = c 2 + ic 4 tend to zero as t::.t tends to zero.
Hence, by (6.9-2),

f ( ) - -i8^1. lmLli.t->0 'Xt Lli.w

e z - e lz'(t)I

-i8 [c. ) x'(t) (. ) y'(t) ]

= e Ux + Wx lz'(t)I + Uy+ ivy lz'(t)I

= e-i^8 [( Ux + ivx) cos 0 + (Uy +ivy) sin OJ

By using (6.7-11) and (6.7-12) in (6.10-1) we find that

f~(z) = e-i^8 [(fz + fz) cos 0 + i(f z - fz) sin OJ

= e-i^8 [f z( cos (J + i sin 0) + fz( cos 0 - i sin O)]

= f z + fze-^2 i^8

(6.10-1)

(6.10-2)
This formula (with a different notation) was first given by B. Riemann
in his doctoral dissertation [101]. E. Kasner ([66] and [68]) has called fz

the mean derivative of f at z, while fz is called the phase derivative of

f at z. The value f z is also called the areolar or areal derivative of f at

z. The concept of the areolar derivative, introduced by Pompeiu in 1912