Bürmann's theorem deals with the expansion of functions in powers of another function. Let 
be a function of 
which is analytic in a closed region 
, of which 
is an interior point, and let 
. Suppose also that 
. Then Taylor's theorem
gives the expansion
 |
(1)
|
and, if it is legitimate to revert this series, the expression
![z-a=(phi(z)-b)/(phi^'(a))-1/2(phi^('')(a))/([phi^'(a)]^3)[phi(z)-b]^2+...](/images/equations/BuermannsTheorem/NumberedEquation2.svg) |
(2)
|
is obtained which expresses 
as an analytic function
of the variable 
for sufficiently small values of 
. If 
is then analytic near 
, it follows that 
is an analytic function
of 
when 
is sufficiently small, and so there will be an expansion in the form
![f(z)=f(a)+a_1[phi(z)-b]+(a_2)/(2!)[phi(z)-b]^2+(a_3)/(3!)[phi(z)-b]^3+...](/images/equations/BuermannsTheorem/NumberedEquation3.svg) |
(3)
|
(Whittaker and Watson 1990, p. 129).
The actual coefficients in the expansion are given by the following theorem, generally known as Bürmann's theorem (Whittaker and Watson 1990, p. 129). Let 
be a function of 
defined by the equation
 |
(4)
|
Then an analytic function 
can, in a certain domain of values of 
, be expanded in the form
![f(z)=f(a)+sum_(m=1)^(n-1)([phi(z)-b]^m)/(m!)(d^(m-1))/(da^(m-1)){f^'(a)[psi(a)]^m}+R_n,](/images/equations/BuermannsTheorem/NumberedEquation5.svg) |
(5)
|
where the remainder term is
![R_n=1/(2pii)int_a^zint_gamma[(phi(z)-b)/(phi(t)-b)]^(n-1)(f^'(t)phi^'(z)dtdz)/(phi(t)-phi(z)),](/images/equations/BuermannsTheorem/NumberedEquation6.svg) |
(6)
|
and 
is a contour in the 
-plane enclosing the points 
and 
such that if 
is any point inside 
, the equation 
has no roots on or inside the contour
except a simple root 
(Whittaker and Watson 1990, p. 129).
Teixeira's theorem is an extended form of Bürmann's theorem. The Lagrange inversion theorem gives another such extension.
