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]= |  |
