Lagrange Series Inversion
Recover an inverse branch from Catalan coefficients and compare its finite series with the exact quadratic inverse.
Free theory articleRecover an inverse branch from Catalan coefficients and compare its finite series with the exact quadratic inverse.
Concept & interpretation
The Lagrange inversion theorem constructs the Taylor expansion of the inverse branch of an analytic map around a point where the first derivative is nonzero. For the quadratic equation y = x + cy2 (equivalently forward map x = y - cy2), the branch passing through the origin y(0) = 0 has exact closed form y(x) = 2x / (1 + √(1 - 4cx)) and Catalan series expansion:
Interactive Experiment
Explore Catalan convergence and branch-point singularities in the Lagrange Series Inversion interactive workspace:
- Verify the default state (Coefficient c = 1, Probe x = 1/8, Degree N = 3) displays Truncated inverse = 37/256, Series residual = -89/65536, and Inverse derivative = 1.4142135623731.
- Flip Coefficient c = -1 (keeping Probe x = 1/8, Degree N = 3). Alternating signs yield
y3 = 1/8 - 1/64 + 1/256 = 29/256with positive residual73/65536. - Set Coefficient c = 0 and Probe x = -1/4. All higher Catalan terms vanish and the linear branch
y = x = -1/4is reproduced with zero residual and zero error. - Set Coefficient c = 1, Probe x = 1/4, and Degree N = 6. At the branch endpoint
1 - 4cx = 0, the analytic value stays finite (y = 0.5), whereas Inverse derivative becomes Unavailable (Unbounded at branch endpoint).