A power series which offers a complete solution to the
problem of local inversion of holomorphic functions. In fact, let a function
of the complex variable
be regular in a neighbourhood of the point
,
and let
and
.
Then there exists a regular function
in some neighbourhood of the point
of the
-plane
which is the inverse to
and is such that
.
Moreover, if
is any regular function in a neighbourhood of the point
,
then the composite function
can be expanded in a
Bürmann–Lagrange series
in a neighbourhood of the point
The inverse of the function
is obtained by setting
.
The expansion
(*)
follows from
Bürmann's theorem
[1]:
Under the assumptions made above on the holomorphic functions
and
,
the latter function may be represented in a certain domain in the
-plane
containing
in the form
where
Here
is a contour in the
-plane
which encloses the points
and
,
and is such that if
is any point inside
,
then the equation
has no roots on
or inside
other than the simple root
.
The expansion
(*)
for the case
was obtained by
J.L. Lagrange
.
If the derivative
has a zero of order
at the point
,
there is the following generalization of the
Bürmann–Lagrange series for the multi-valued inverse function
[3]:
Another generalization (see, for example,
[4])
refers to functions
regular in an annulus; instead of the series
(*),
one obtains
a series with positive and negative powers of the difference
.