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

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) |

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

(11) |

where

(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.

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1999-05-26