3.6.4 Advanced Topic: 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 .

In[1]:= Series[Exp[x], {x, 0, 5}]

Out[1]=

This replaces the variable in the power series for by a power series for .

In[2]:= ComposeSeries[%, Series[Sin[x], {x, 0, 5}]]

Out[2]=

The result is the power series for .

In[3]:= Series[Exp[Sin[x]], {x, 0, 5}]

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 .

In[4]:= Series[Sin[y], {y, 0, 5}]

Out[4]=

Inverting the series gives the series for .

In[5]:= InverseSeries[%, x]

Out[5]=

Composing the two series gives the identity function.

In[6]:= ComposeSeries[%, %%]

Out[6]=