1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 313

the point wo has only one inverse image). But D.w ~ 0 as D.z ~ 0, and

conversely. Choosing z E N 0 1(zo) we have

or

f(z) - f(zo)

---'--'---'--'-=

w-wo

Z -Zo F(w)-F(wo)

F(w) - F(wo) =

1

+ J(z) - f(zo)

W-Wo z - Zo

and letting w ~ w 0 we get F'(wo) = 1/ f'(zo).

6.4 Differentiability of a Real Function of Two Real Variables

We recall the classical definition (Thomae-Stolz): Let u = u(x, y) be a real

function defined in some open set D of the xy-plane. Let (x, y) be some

fixed point in D, and (x + D.x, y +D.y) an arbitrary point in N 0 1(x, y) CD.

The function u(x, y) is said to be differentiable at (x, y) iff the increment

D.u = u(x + D.x,y + D.y) - u(x,y) can be expressed in the form

D.u = A D.x + B D.y + c1 D.x + c2 D.y (6.4-1)

where A and B depend on the point (x, y), but not on the increments D.x,

D.y, and c 1 , c 2 are functions of D.x and D.y such that c 1 ~ 0 and c2 ~ 0

as D.x ~ 0, D.y ~ 0.

By taking D.x f O, D.y = 0 in (6.4-1), it follows that

au. D.u

Ux= - = hm - =A

ax .t..x-;+0 D.x

and similarly, Uy = B. Thus a necessary condition for u to be differentiable

at a point is the existence of ilie_12artial derivatives u., ~ud Uy at that n.oint.

'Since Ux =A, Uy = B, (6.4-1) can always be written in the form

D.u = Ux D.x +Uy D.y + c (6.4-2)

where c = c 1 D.x+c: 2 D.y. The sum of the first two terms in (6.4-2) is called

the principal part of the increment D.u, and also the total differential of u,

or simply, the differential of u, and we write

du= Uxdx+uydy

with the further conventions dx = D.x and dy = D.y. As to the last term,

c:, it is easy to see that it represents an infinitesimal of higher order with

respect to ID.zl = J D.x^2 + D.y^2 , or with respect to D.z = D.x + i D.y. In