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BUILT-IN MATHEMATICA SYMBOL

InverseSeries

InverseSeries[s]
takes the series s, and gives a series for the inverse of the function represented by s.

uses the variable x in the inverse series.

InverseSeries can be applied to any SeriesData object with the appropriate structure, whether or not it has been generated by Series.

Find a series for the inverse function of x Sin[x]:

Find a series for the inverse function of Sin

Find the series for the inverse of the function represented by an explicitly specified series:

Invert again to get back the original function:

Find a series for the inverse function of x Sin[x]:

Out[1]=

Find a series for the inverse function of Sin

Out[1]=

Out[2]=

Find the series for the inverse of the function represented by an explicitly specified series:

Out[1]=

Invert again to get back the original function:

Out[2]=

Invert a series, giving a result in terms of the variable

Find higher-order terms in Newton's approximation for a root of

near

Composing an inverse series with the original series gives the identity function: