Series reversion is the computation of the coefficients of the inverse function given those of the forward function. For a function expressed in a series with no constant term (i.e., ) as

(1)

the series expansion of the inverse series is given by

(2)

By plugging (2) into (1), the following equation is obtained

(3)

Equating coefficients then gives

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

(Dwight 1961, Abramowitz and Stegun 1972, p. 16).

Series reversion is implemented in Mathematica as InverseSeries[s, x], where is given as a SeriesData object. For example, to obtain the terms shown above,

  With[{n = 7},
    CoefficientList[
      InverseSeries[SeriesData[x, 0, Array[a, n], 1, n + 1, 1]],
    x]
  ]

A derivation of the explicit formula for the th term is given by Morse and Feshbach (1953),

(11)

where

(12)

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 316-317, 1985.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 297, 1987.

Dwight, H. B. Table of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 411-413, 1953.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 22, 1995.

CITE THIS AS:

Weisstein, Eric W. "Series Reversion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SeriesReversion.html