6.36.3  The inverse of a series: revert

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

• Input:
  revert (x + x^2 + x^4)
  Output:

  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 x − x² + 2x³ − 6x⁴.
• Input:
  revert(1 + x + x^2/2 + x^3/6 + x^4/24)
  Note that the argument is the beginning of the power series for exp(x), so the output is the beginning of the power series for ln(1+x). Output:

  x−

  x2 2

  +

  x3 3

  −

  x4 4