### 13.5.3 Inverse of a series

The revert command finds the beginning of the power series of the inverse function given the beginning of the series of the original function.

- revert takes one mandatory and one optional argument:
  - series, the beginning of a power series centered at 0 for a function f.
  - Optionally x, the name of the variable (by default x).
- revert(series ⟨,x ⟩) returns the beginning of the power series for the inverse of f; namely the beginning of the power series for g(f(0)+x) where the function g satisfies g(f(x))=x.

#### Examples

Find the series expansion of f−1(x) where f(x)=x+x2+x4+⋯

revert(x+x^2+x^4)

x−x2+2 x3−6 x4 |  |  |  |  |  |  |  |  |  |

Note that if the power series of a function f begins with x+x2+x4, then f(0)=0, f′(0)=1, f′′(0)=2, f′′′(0)=0 and f(4)(0)=24. The function g with g(f(x))=x will then satisfy g(0)=0, g′(0)=1/f′(0)=1, g′′(0)=−2, g′′′(0)=12 and g(4)(0)=−144. The power series for g will then begin with x−x2+2x3−6x4.

If the argument is the beginning of the power series for ex, then the output will be the beginning of the power series for ln(1+x).

revert(1+x+x^2/2+x^3/6+x^4/24)

x− |
x2
2
| + |
x3
3
| − |
x4
4