Lagrange Series Inversion
Recover an inverse branch from Catalan coefficients and compare its finite series with the exact quadratic inverse.
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Lagrange Series Inversion
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:
yN(x) = Σj=0N-1 Cj cj xj+1, where Cj = (2j)! / ((j + 1)! j!) are the Catalan numbers 1, 1, 2, 5, 14, 42, ....
Worked Example: Positive Quadratic Map at c = 1, x = 1/8, N = 3
Compare the 3-term Catalan series approximation against the analytic branch for c = 1, x = 1/8, and N = 3:
- Analytic inverse branch:
y = 2(1/8) / (1 + √(1 - 4(1)(1/8))) = (1/4) / (1 + 1/√2) = (2 - √2)/4 ≈ 0.14644660940672627, with derivativedy/dx = 1 / √(1 - 4cx) = √2 ≈ 1.4142135623730951(Finite). - Truncated Catalan series (
j = 0, 1, 2):y3 = C0(1/8) + C1(1/8)2 + C2(1/8)3 = 1/8 + 1/64 + 2/512 = 37/256 = 0.14453125. - Series residual and error: Substituting
y3 = 37/256intoy3 - x - cy32gives37/256 - 1/8 - (37/256)2 = 5/256 - 1369/65536 = -89/65536, and the absolute approximation error is|y - y3| ≈ 0.0019153594067262.
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).
Limitations and Assumptions
The Lagrange power series for y = x + cy2 has radius of convergence |x| = 1 / (4|c|), set by the fold singularity where dx/dy = 1 - 2cy = 0. At the boundary 4cx = 1, the inverse value y = 1/(2c) is finite but the square-root branch has a vertical tangent (unbounded derivative 1/√(1 - 4cx)) and algebraic O(N-1/2) convergence. Away from c = 0 or x = 0, no finite truncation yN is an exact algebraic inverse.
References
Power series reversion and the Lagrange inversion formula for analytic functions are documented in the following reference:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Lagrange Series Inversion module.
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