# 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

to order

.

Out[1]= | |

This replaces the variable

in the power series for

by a power series for

.

Out[2]= | |

The result is the power series for

.

Out[3]= | |

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

.

Out[4]= | |

Inverting the series gives the series for

.

Out[5]= | |

This agrees with the direct series for

.

Out[6]= | |

Composing the series with its inverse gives the identity function.

Out[7]= | |