292
III - Convergence: Continuous variables
N ow for a matrix
to be invertible, it is necessary and sufficient that its determinant ad - be be
=1= O. In the present case, i.e. of the Jacobian matrix (19.4) of f, this is the
expression D11t(z)D2h(z) - D2!1(z)D1h(z); one calls this the Jacobian of
the map f at the point z = (x, y), from the name of its inventor (formula for
change of variables in a multiple integral), notation Jf(z), or the functional
determinant of the pair of functions It, 12, denoted classically by
(24.7)
Suppose for example that It + ih = f is holomorphic. Cauchy's relations
signify precisely that the second matrix (6) is of the form
(24.8)
so that (7) is
(24.9)
Since the inverse of (8) is
( a' b')
, 2 2
-b' a' where a = aj(a + b), b' = -bj(a 2 + b 2 ),
the first matrix figuring in the left hand side of (6) is then also of the form
(8), which means that, if it exists, the function g1 + ig2 satisfies Cauchy's
conditions, i.e. is holomorphic like f. Inverting a holomorphic function leads
to a holomorphic function if what we hope is indeed correct.
After these explanations, we shall first examine the problem of local inversion - it will provide the solution of the other almost gratis, as we shall
see - and establish the following result:
Theorem 24 ("local inversion"). Let G be an open subset of]R2 and let f
be a map of class C l ofG into ]R2. Suppose that the derivative map Df(c) is
invertible at a point c E G. Then there is an open U c G, containing c, which
f maps bijectively onto an open V and such that the inverse map g : V ~ U
is of class C l on V.
III - Convergence: Continuous variables
N ow for a matrix
to be invertible, it is necessary and sufficient that its determinant ad - be be
=1= O. In the present case, i.e. of the Jacobian matrix (19.4) of f, this is the
expression D11t(z)D2h(z) - D2!1(z)D1h(z); one calls this the Jacobian of
the map f at the point z = (x, y), from the name of its inventor (formula for
change of variables in a multiple integral), notation Jf(z), or the functional
determinant of the pair of functions It, 12, denoted classically by
(24.7)
Suppose for example that It + ih = f is holomorphic. Cauchy's relations
signify precisely that the second matrix (6) is of the form
(24.8)
so that (7) is
(24.9)
Since the inverse of (8) is
( a' b')
, 2 2
-b' a' where a = aj(a + b), b' = -bj(a 2 + b 2 ),
the first matrix figuring in the left hand side of (6) is then also of the form
(8), which means that, if it exists, the function g1 + ig2 satisfies Cauchy's
conditions, i.e. is holomorphic like f. Inverting a holomorphic function leads
to a holomorphic function if what we hope is indeed correct.
After these explanations, we shall first examine the problem of local inversion - it will provide the solution of the other almost gratis, as we shall
see - and establish the following result:
Theorem 24 ("local inversion"). Let G be an open subset of]R2 and let f
be a map of class C l ofG into ]R2. Suppose that the derivative map Df(c) is
invertible at a point c E G. Then there is an open U c G, containing c, which
f maps bijectively onto an open V and such that the inverse map g : V ~ U
is of class C l on V.