12.5.3 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+⋯
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) |