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
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(1)
|
the series expansion of the inverse series is given by
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(2)
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By plugging (2) into (1), the following equation is obtained
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(3)
|
Equating coefficients then gives
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
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(10)
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(Dwight 1961, Abramowitz and Stegun 1972, p. 16).
Series reversion is implemented in the Wolfram Language 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),
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(11)
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where
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(12)
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