Series Reversion

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 as

$$y=a_1x+a_2x^2+a_3x^3+\ldots,$$ (1)

the series expansion of the inverse series is given by

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

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

$y=a_1A_1y+(a_2{A_1}^2+a_1A_2)y^2$
$+(a_3{A_1}^3+2a_2A_1A_2+a_1A_3)y^3$
$+(3a_3{A_1}^2A_2+a_2{A_2}^2+a_2A_1A_3)+\ldots.\quad$	(3)

Equating Coefficients then gives

$$A_1 = {a_1}^{-1}$$ (4)

$$A_2 = -{a_2\over a_1}{A_1}^2 = -{a_1}^{-3}a_2$$ (5)

$$A_3 = {a_1}^{-5}(2{a_2}^2-a_1a_3)$$ (6)

$$A_4 = {a_1}^{-7}(5a_1a_2a_3-{a_1}^2a_4-5{a_2}^3)$$ (7)

$$A_5 = {a_1}^{-9}(6{a_1}^2a_2a_4+3{a_1}^2a_2a_3+14{a_2}^4-{a_1}^3a_5-21a_1{a_2}^2a_3)$$ (8)

$$A_6 = {a_1}^{-11}(7{a_1}^3a_2a_5+7{a_1}^3a_3a_4+84{a_1}{a_2}^3{a_3}$$

$$\phantom{=} -{a_1}^4{a_6}-28{a_1}^2{a_2}{a_3}^2-42{a_2}^5-28{a_1}^2{a_2}^2{a_4})$$ (9)

$$A_7 = {a_1}^{-13}(8{a_1}^4{a_2}{a_6}+8{a_1}^4{a_3}{a_5}+4{a_1}^4{a_4}^2$$

$$\phantom{=} +120{a_1}^2{a_2}^3{a_4}+180{a_1}^2{a_2}^2{a_3}^2+132{a_2}^6$$

$$\phantom{=} -{a_1}^5a_7-36{a_1}^3{a_2}^2{a_5}-72{a_1}^3{a_2}{a_3}{a_4}-12{a_1}^3{a_3}^3-330{a_1}{a_2}^4{a_3})$$ (10)

(Dwight 1961, Abramowitz and Stegun 1972, p. 16). A derivation of the explicit formula for the $n$th term is given by Morse and Feshbach (1953),

$A_n={1\over n{a_1}^n} \sum_{s, t, u,\ldots} (-1)^{s+t+u+\ldots} {n(n+1)\cdots(n-1+s+t+u+\ldots)\over s!t!u!\cdots}$
$\times \left({a_2\over a_1}\right)^s\left({a_3\over a_1}\right)^t\cdots,\quad$	(11)

where

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

References

Abramowitz, M. and Stegun, C. 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.