§5. Differentiable functions of several variables
287
Hence the passage to the limit which transforms (2) into (3).
We leave it to the reader to extend this limit theorem for derivatives to
the case of arbitrary functions of class CPo
Finally we examine the case of holomorphic functions. If the fn are holomorphic in U, it suffices to assume compact convergence of the complex
derivatives f~ to a limit g. It is clear that Cauchy's formulae then also apply
to the limit f of fn, which is thus holomorphic and satisfies l' = g. Despite
appearances, this result is of no interest, because, when dealing with holomorphic or analytic functions, the compact convergence of f n (and even much
less) implies that of the derivatives, as we shall see in Chap. VII; one uses this
"miraculous" result constantly. It generalises, not without a little more work
- say a century to obtain the general case starting from the particular case
of holomorphic or harmonic functions - to the solutions of the much larger
class of "elliptic" linear partial differential equations of arbitrary order.
In all cases, it is clear that if the f n satisfy a partial differential equation
of some reasonable kind, for example
(f" i")2 f'
.
I
xx -
yy
+ x = smx + ogy,
and if the derivatives of order ~ 2 converge, then the limit function again
satisfies the same equation. But in a case of this kind, the convergence of
solutions does not imply that of their derivatives nor that the limit is again a
solution, unless one adopts the point of view of Schwartz' theory of distributions, an artifice which, despite its usefulness, neither permits one to prove
false theorems nor exempts one from proving true theorems.
23 - Interchanging the order of differentiation
Consider a holomorphic function f, and assume that it is of class C 2 - hardly
a restrictive hypothesis since we shall learn that f is analytic and so Coo -,
so that its derivative l' is of class Cl. If f is analytic, so also is 1', and it
seems to follow that l' is holomorphic. Might one prove this directly from
Cauchy's relation?
Since l' = Dl f = -iDd and so
Cauchy'S relation Dd' = -iDd' for l' reduces to DIDd = D2Dd. It is
also satisfied trivially by the nonholomorphic function xPyq, for example.
Hence arises the general problem of interchanging the order of differentiation
(23.1)
where, of course, one no longer supposes f holomorphic. To establish this
fundamental relation when it is true, assume for a start that f is of class C 1
287
Hence the passage to the limit which transforms (2) into (3).
We leave it to the reader to extend this limit theorem for derivatives to
the case of arbitrary functions of class CPo
Finally we examine the case of holomorphic functions. If the fn are holomorphic in U, it suffices to assume compact convergence of the complex
derivatives f~ to a limit g. It is clear that Cauchy's formulae then also apply
to the limit f of fn, which is thus holomorphic and satisfies l' = g. Despite
appearances, this result is of no interest, because, when dealing with holomorphic or analytic functions, the compact convergence of f n (and even much
less) implies that of the derivatives, as we shall see in Chap. VII; one uses this
"miraculous" result constantly. It generalises, not without a little more work
- say a century to obtain the general case starting from the particular case
of holomorphic or harmonic functions - to the solutions of the much larger
class of "elliptic" linear partial differential equations of arbitrary order.
In all cases, it is clear that if the f n satisfy a partial differential equation
of some reasonable kind, for example
(f" i")2 f'
.
I
xx -
yy
+ x = smx + ogy,
and if the derivatives of order ~ 2 converge, then the limit function again
satisfies the same equation. But in a case of this kind, the convergence of
solutions does not imply that of their derivatives nor that the limit is again a
solution, unless one adopts the point of view of Schwartz' theory of distributions, an artifice which, despite its usefulness, neither permits one to prove
false theorems nor exempts one from proving true theorems.
23 - Interchanging the order of differentiation
Consider a holomorphic function f, and assume that it is of class C 2 - hardly
a restrictive hypothesis since we shall learn that f is analytic and so Coo -,
so that its derivative l' is of class Cl. If f is analytic, so also is 1', and it
seems to follow that l' is holomorphic. Might one prove this directly from
Cauchy's relation?
Since l' = Dl f = -iDd and so
Cauchy'S relation Dd' = -iDd' for l' reduces to DIDd = D2Dd. It is
also satisfied trivially by the nonholomorphic function xPyq, for example.
Hence arises the general problem of interchanging the order of differentiation
(23.1)
where, of course, one no longer supposes f holomorphic. To establish this
fundamental relation when it is true, assume for a start that f is of class C 1