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Series Reversion

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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., a_0=0) as

y=a_1x+a_2x^2+a_3x^3+...,
(1)

the series expansion of the inverse series is given by

x=A_1y+A_2y^2+A_3y^3+....
(2)

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

y=a_1A_1y+(a_2A_1^2+a_1A_2)y^2+(a_3A_1^3+2a_2A_1A_2+a_1A_3)y^3+(3a_3A_1^2A_2+a_2A_2^2+a_2A_1A_3)+....
(3)

Equating coefficients then gives

A_1=a_1^(-1)
(4)
A_2=-a_1^(-3)a_2
(5)
A_3=a_1^(-5)(2a_2^2-a_1a_3)
(6)
A_4=a_1^(-7)(5a_1a_2a_3-a_1^2a_4-5a_2^3)
(7)
A_5=a_1^(-9)(6a_1^2a_2a_4+3a_1^2a_3^2+14a_2^4-a_1^3a_5-21a_1a_2^2a_3)
(8)
A_6=a_1^(-11)(7a_1^3a_2a_5+7a_1^3a_3a_4+84a_1a_2^3a_3-a_1^4a_6-28a_1^2a_2a_3^2-42a_2^5-28a_1^2a_2^2a_4)
(9)
A_7=a_1^(-13)(8a_1^4a_2a_6+8a_1^4a_3a_5+4a_1^4a_4^2+120a_1^2a_2^3a_4+180a_1^2a_2^2a_3^2+132a_2^6-a_1^5a_7-36a_1^3a_2^2a_5-72a_1^3a_2a_3a_4-12a_1^3a_3^3-330a_1a_2^4a_3)
(10)

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

Series reversion is implemented in the Wolfram Language as InverseSeries[s, x], where s 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 nth term is given by Morse and Feshbach (1953),

A_n=1/(na_1^n)sum_(s,t,u,...)(-1)^(s+t+u+...)(n(n+1)...(n-1+s+t+u+...))/(s!t!u!...)((a_2)/(a_1))^s((a_3)/(a_1))^t...,
(11)

where

s+2t+3u+...=n-1.
(12)

See also

Power Series, Series

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References

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.

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Series Reversion

Cite this as:

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

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