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 exp(x) to order (x^5):

Series[Exp[x],{x,0,5}]

This replaces the variable x in the power series for exp(x) by a power series for sin(x):

ComposeSeries[%,Series[Sin[x],{x,0,5}]]

The result is the power series for exp(sin(x)):

Series[Exp[Sin[x]],{x,0,5}]

If you have a power series for a function f(y), then it is often possible to get a power series approximation to the solution for y in the equation f(y)==x. This power series effectively gives the inverse function (f^(-1))(x) such that f((f^(-1))(x))==x. The operation of finding the power series for an inverse function is sometimes known as reversion of power series.

Here is the series for sin(y):

Series[Sin[y],{y,0,5}]

Inverting the series gives the series for (sin^(-1))(x):

InverseSeries[%,x]

This agrees with the direct series for (sin^(-1))(x):

Series[ArcSin[x],{x,0,5}]

Composing the series with its inverse gives the identity function:

ComposeSeries[%,%%%]