1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 331

(6.8-11)

Similarly,

af af aw1 af aw2.

-=--+--

ay aw1 ay aw2 ay

(6.8-12)

Therefore, the usual rules of real analysis for the derivatives of composite

functions are also valid in the complex case, provided that the function

f( wi, w2) is analytic with respect to w1 and w 2 separately. Next, we have

azJ(w1,w2) = 2 1 (af ax - i .af) ay

and making use of (6.8-11) and (6.8-12), we get

azf(wi,w 2 ) = ~ [( af aw1 + af aw2)

2 aw1 ax aw2 ax

_ i ( ~ aw1 + ~ aw2 ) ]

aw1 ay aw2 ay

= a f. ~ ( aw1 - i aw1 ) + a f. ~ ( aw2 -i aw2 )

aw1 2 ax ay aw2 2 ax ay

af af

= -a W1 azWl + -a W2 azW2

which is formula (6.8-9). Similarly, formula (6.8-10) is established.

Corollary 6.1 We have

- af

azf(z, z) = az'

- af

azf(z, z) = a:z (6.8-13)

Proof It suffices to choose w 1 = z, w 2 = z in (6.8-9) and (6.8-10), and

make use of Theorem 6.7.

The corollary above shows that the symbols azf and azf can be

interpreted indeed as the formal partial derivatives af /az and af /az,

respectively, provided that f ( w 1 , w 2 ) is analytic in w 1 and w 2 separately.

EXERCISES 6.1

1. Prove the differentiation rules in Theorem 6.2-2.


2. Find the derivatives of the following functions.

(a) w = zn (n a positive integer) (b) w = z-n