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+⋯

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 e^x, then the output will be the beginning of the power series for ln(1+x).