Composition and Inversion of Power Series
When you manipulate power series, it is sometimes convenient to think of the series as representing functions, which you can, for example, compose or invert.
Composition and inversion of power series.
Here is the power series for
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.