278
III - Convergence: Continuous variables
whence the existence of f' and the first formula (6). The relation df = f'dz
is also immediate since
df = Dd.dx + D2/.dy = Dd(dx + idy) = j'dz,
qed.
This argument in fact shows that f is holomorphic if and only if its
derivative maps in the real sense are C-linear, i.e. of the form
(u, v) 1------+ c(u + iv)
with a constant c E C, and not only lR-linear: this is exactly what Cauchy's
relations express.
We can express this fact in a more striking way by noting that the formulae
(20.7)
dz = dx +idy,
dz = dx - idy
conversely give
(20.8)
dx = (dz + dz)/2,
dy = (dz - dz)/2ij
the differential df = Dddx + D2/dy of every differentiable function f, holomorphic or not, can thus always be put in the form
(20.9)
of
of
df = -dz+ -dz
oz
oz
where by definition
(20.10) of /oz = (Dd - iD2/)/2,
of /oz = (Dd + iD2/)/2j
here these are pure notational conventions, not to be confused with the usual
derivatives defined by passing to the limit in a quotient.
In the formula (9), dz is a C-linear function (u,v) I--t c(u + iv), but dz,
of the form (u, v) I--t d(u - iv), is not. The holomorphic functions are thus
characterised by the relation
of/oz=O,
obviously equivalent to Cauchy's condition by (10). Then
of /oz = /"
the derivative of f in the complex sense.
Since the rules for calculating derivatives established in nO 15 apply to
functions of several variables, with the same proofs for those who care to prove
them, it is clear that the sum, the product and the quotient of two holomorphic
functions are again holomorphic: it is enough to check that these algebraic
operations do not violate Cauchy's condition. For example:
III - Convergence: Continuous variables
whence the existence of f' and the first formula (6). The relation df = f'dz
is also immediate since
df = Dd.dx + D2/.dy = Dd(dx + idy) = j'dz,
qed.
This argument in fact shows that f is holomorphic if and only if its
derivative maps in the real sense are C-linear, i.e. of the form
(u, v) 1------+ c(u + iv)
with a constant c E C, and not only lR-linear: this is exactly what Cauchy's
relations express.
We can express this fact in a more striking way by noting that the formulae
(20.7)
dz = dx +idy,
dz = dx - idy
conversely give
(20.8)
dx = (dz + dz)/2,
dy = (dz - dz)/2ij
the differential df = Dddx + D2/dy of every differentiable function f, holomorphic or not, can thus always be put in the form
(20.9)
of
of
df = -dz+ -dz
oz
oz
where by definition
(20.10) of /oz = (Dd - iD2/)/2,
of /oz = (Dd + iD2/)/2j
here these are pure notational conventions, not to be confused with the usual
derivatives defined by passing to the limit in a quotient.
In the formula (9), dz is a C-linear function (u,v) I--t c(u + iv), but dz,
of the form (u, v) I--t d(u - iv), is not. The holomorphic functions are thus
characterised by the relation
of/oz=O,
obviously equivalent to Cauchy's condition by (10). Then
of /oz = /"
the derivative of f in the complex sense.
Since the rules for calculating derivatives established in nO 15 apply to
functions of several variables, with the same proofs for those who care to prove
them, it is clear that the sum, the product and the quotient of two holomorphic
functions are again holomorphic: it is enough to check that these algebraic
operations do not violate Cauchy's condition. For example: