§5. Differentiable functions of several variables
277
which is o(v) when (u, v) tends to (0,0). In view of (19.4), one finds 41
I(a + u, b + v) = I(a, b) + Dd(a, b + v)u + Dd(a, b)v + o(lul + Ivl);
but since Dd is continuous, one has Dd(a, b + v)u = Dd(a, b)u + o(u). In
conclusion:
Theorem 21. Let I be a function 01 class C 1 on an open subset U 01]R2.
Then I is differentiable at every point (x, y) E U, and
(20.4)
I(x+u,y+v) =
= I(x, y) + Dd(x, y)u + D2/(x, y)v + o(lul + Ivl)
when u and v tend to o.
This result has an immediate consequence in the theory of analytic functions. We saw in Chap. II, nO 19 that if a function I(z) is analytic on an open
U of C, i.e. is expandable in a power series on a neighbourhood of any point
of U, then it admits a derivative
(20.5)
J'(z) = lim I(z + h) - I(z)
h->O
h
in the complex sense where h = u + iv tends to 0 through complex values; and we deduced from this, by an immediate calculation, that I then
has continuous partial derivatives Dd and Dd satisfying Cauchy's identity
Dd = iDd, i.e. that I is holomorphic. We stated that, conversely, every
holomorphic function is analytic. We cannot establish this yet - this will be
one of the aims of Chap. VII - but we can now make a step in the right
direction:
Corollary. Let I be a function 01 class Cion an open subset U 01 c. II I
satisfies Cauchy's equation D21 = iDd (i.e. is holomorphic), then I has a
complex derivative (5) at all points z E U, and
(20.6)
dl = J'(z)dz.
Indeed, the relation (4) can now be written
I(x + u, y + v) = I(x, y) + Dd(x, y)(u + iv) + o(lul + Ivl)
or, on putting h = u + iv and z = x + iy,
I(z + h) = I(z) + Dd(z)h + o(h),
41 Subject to showing that o(u) + o(v) = o(lul + Iv!), which is clear, since, for all
r > 0, the expressions on the left hand side are, for lui + Ivl small, majorised by
rlul and rlvl respectively.