Bürmann's Theorem

Bürmann's theorem deals with the expansion of functions in powers of another function. Let phi(z) be a function of z which is analytic in a closed region S, of which a is an interior point, and let phi(a)=b. Suppose also that phi^'(a)!=0. Then Taylor's theorem gives the expansion

and, if it is legitimate to revert this series, the expression

is obtained which expresses z as an analytic function of the variable phi(z)-b for sufficiently small values of |z-a|. If f(z) is then analytic near z=a, it follows that f(z) is an analytic function of phi(z)-b when |z-a| is sufficiently small, and so there will be an expansion in the form

(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 psi(z) be a function of z defined by the equation

Then an analytic function f(z) can, in a certain domain of values of z, be expanded in the form

where the remainder term is

and gamma is a contour in the t-plane enclosing the points a and z such that if zeta is any point inside gamma, the equation phi(t)=phi(zeta) has no roots on or inside the contour except a simple root t=zeta (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.

See also

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References

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Cite this as:

Weisstein, Eric W. "Bürmann's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BuermannsTheorem.html

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