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GeezerVison is showing mathworld.wolfram.com — Bürmann's Theorem -- from Wolfram MathWorld

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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

phi(z)-b=phi^'(a)(z-a)+(phi^('')(a))/(2!)(z-a)^2+...,
(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+...
(2)

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

f(z)=f(a)+a_1[phi(z)-b]+(a_2)/(2!)[phi(z)-b]^2+(a_3)/(3!)[phi(z)-b]^3+...
(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 psi(z) be a function of z defined by the equation

psi(z)=(z-a)/(phi(z)-b).
(4)

Then an analytic function f(z) can, in a certain domain of values of z, 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,
(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)),
(6)

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

Darboux's Formula, Lagrange Inversion Theorem, Taylor Series, Teixeira's Theorem

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References

Dixon, A. C. "On Burmann's Theorem." Proc. London Math. Soc. 34, 151-153, 1902.Lagrange, J.-L. and Legendre, A. M. "Rapport sur deux mémoires d'analyse du professeur Burmann." Mémoires de l'Institut National des Sci. et Arts: Sci. Math. Phys. 2, 13-17, 1799.Whittaker, E. T. and Watson, G. N. "Bürmann's Theorem" and "Teixeira's Extended Form of Bürmann's Theorem." §7.3 and 7.3.1 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 128-132, 1990.

Referenced on Wolfram|Alpha

Bürmann's Theorem

Cite this as:

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

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