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Teixeira's Theorem

An extended form of Bürmann's theorem. Let f(z) be a function of z analytic in a ring-shaped region A, bounded by another curve C and an inner curve c. Let theta(z) be a function analytic on and inside C having only one zero a (which is simple) within the contour. Further let x be a given point within A. Finally, let

|theta(x)|<|theta(z)|
(1)

for all points z of C, and

|theta(x)|>|theta(z)|
(2)

for all points z of c. Then

f(x)=sum_(n=0)^inftyA_n[theta(x)]^n+sum_(n=1)^infty(B_n)/([theta(x)]^n),
(3)

where

A_n=1/(2pii)int_C(f(z)theta^'(z)dz)/([theta(z)]^(n+1))
(4)
B_n=1/(2pii)int_cf(z)[theta(z)]^(n-1)theta^'(z)dz
(5)

(Whittaker and Watson 1990, pp. 131-132).

See also

Bürmann's Theorem, Lagrange Inversion Theorem

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References

Bateman, H. "An Extension of Lagrange's Expansion." Trans. Amer. Math. Soc. 28, 346-356, 1926.Teixeira, F. G. "Sur les séries ordonnées suivant les puissance d'une fonction donnée." J. für Math. 122, 97-123, 1900.Whittaker, E. T. and Watson, G. N. "Teixeira's Extended Form of Bürmann's Theorem." §7.31 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 131-132, 1990.

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Teixeira's Theorem

Cite this as:

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

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