This replaces the variable in the power series for by a power series for :
The result is the power series for :
If you have a power series for a function , then it is often possible to get a power series approximation to the solution for in the equation . This power series effectively gives the inverse function such that . The operation of finding the power series for an inverse function is sometimes known as reversion of power series.
Here is the series for :
Inverting the series gives the series for :
This agrees with the direct series for :
Composing the series with its inverse gives the identity function: