Abstract
We develop series and analytic-function lifts to symbolic numerals (m, a). Power series act on the magnitude while carrying alignment unchanged, with convergence, differentiation, and integration inherited from the classical series on m. Uniform convergence guarantees safe interchange with collapse phi(m,a) = m. Examples include exp, log, sin, and cos, with practical remainder bounds and boundary guards.
Power series on the magnitude channel
Let
f(m) = sum_{n=0}^inf c_n * (m - m0)^n
have radius of convergence R in (0, +inf]. Define the symbolic lift on the disc
D = { m : |m - m0| < R }:
f*(m, a) = ( f(m) , a )
- Alignment carry: unary series act on
mand preserve the input alignmenta. - Collapse:
phi( f*(m,a) ) = f(m)form in D. - Uniform convergence on compact subsets: For any
0 < r < R, the series converges uniformly on|m - m0| <= r, so limits/sums commute with the lift and with collapse.
Ratio/root tests (classical ⇒ lifted).
If limsup_{n→inf} |c_n|^{1/n} = 1/R, the lifted series is valid exactly on |m - m0| < R. Nothing further is required for alignment.
Termwise differentiation and integration
For |m - m0| < R:
(d/dm) f*(m, a) = ( f'(m) , a )
∫ f*(m, a) dm = ( ∫ f(m) dm , a ) + C_s
where C_s = (C, +1) is the integration constant (default alignment +1 unless tagged). Classical theorems (Cauchy–Hadamard, termwise operations inside |m - m0| < R) carry over because the series acts only on m.
Interchange with collapse (safe limits)
Let S_N(m) = sum_{n=0}^N c_n (m - m0)^n. On every compact K ⊂ D,
f*(m,a) = ( lim_{N→inf} S_N(m) , a )
phi( f*(m,a) ) = lim_{N→inf} S_N(m)
Uniform convergence on K justifies:
phi( lim_N S_N*(m,a) ) = lim_N phi( S_N*(m,a) )
So classical limit manipulations remain valid in the symbolic setting.
Error control (remainder bounds)
If f is analytic on |m - m0| <= r < R, the usual bounds apply:
- Taylor remainder (Lagrange form):
|R_N(m)| <= M * |m - m0|^(N+1) / (N+1)!,withM = max_{|z - m0| = r} |f^{(N+1)}(z)|. - Geometric bound: if
| (m - m0) / ρ | = q < 1and|c_n| <= C / ρ^n, then|R_N(m)| <= (C * q^(N+1)) / (1 - q).
Alignment is unaffected; report a unchanged.
Boundary and guard policies
At |m - m0| = R, convergence may fail or become conditional. Use:
series_guard = "error" # default: disallow evaluation on/beyond boundary
| "warn" # allow, but mark as unstable
| "sum_by_method:<name>" # e.g., Abel, Cesàro (declare)
When a resummation is declared, it applies to the magnitude only.
Canonical examples
1) Exponential (entire).
exp*(m, a) = ( sum_{n=0}^inf m^n / n! , a ), R = +inf
- Truncation:
exp(m) ≈ sum_{n=0}^N m^n/n!with remainder bound by the next term timese^{|m|}. - Worked example:
x = (1.0, 0.6)N=4: exp(1) ≈ 1 + 1 + 1/2 + 1/6 + 1/24 = 2.7083 exp*(x) ≈ (2.7083, 0.6) (true: 2.7183..., error ~ 0.0100)
2) Logarithm near 1 (principal real branch).
log(1+m) = sum_{n=1}^inf (-1)^{n+1} * m^n / n, |m| < 1
log1p*(m, a) = ( log(1+m) , a )
- Domain (real principal): require
1+m > 0if using closed form; for the series, require|m| < 1. - Worked example:
x = (0.2, -0.4)log1p*(x) ≈ ( 0.2 - 0.2^2/2 + 0.2^3/3 - 0.2^4/4 , -0.4 ) ≈ ( 0.2 - 0.02 + 0.0026667 - 0.0004 , -0.4 ) ≈ ( 0.182267 , -0.4 )
3) Sine and cosine (entire).
sin*(m,a) = ( sum_{k=0}^inf (-1)^k * m^(2k+1)/(2k+1)! , a )
cos*(m,a) = ( sum_{k=0}^inf (-1)^k * m^(2k) /(2k)! , a )
- Order/odd-even: magnitude properties match classical; alignment is carried.
- Small-angle: for
|m| << 1,sin*(m,a) ≈ (m, a),cos*(m,a) ≈ (1 - m^2/2, a).
Analytic continuation (sketch for real line usage)
If f admits analytic continuation beyond |m - m0| < R, we keep the real principal choice for the magnitude and carry a. If multiple real branches exist (e.g., log, inverse trig/hyperbolic), declare:
branch = <principal | named branch>
branch_guard = "error" | "switch" (explicit switch required)
This mirrors the policies used in 2.31–2.34.
Composition and products of series
If f(m) = sum c_n (m - m0)^n and g(m) = sum d_n (m - m0)^n converge on |m - m0| < R, then on any r < R:
(f + g)*(m,a) = ( f(m) + g(m) , a )
(f * g)*(m,a) = ( (f·g)(m) , a )
For composition, if g(D) ⊂ E and f is analytic on E, the lifted composition
(f ∘ g)*(m,a) = ( f( g(m) ) , a )
is valid on the classical domain (cross-ref 2.38).
Worked mini-case (Chebyshev/Taylor side-by-side)
Approximate cos(m) on [-1,1] with degree 4:
Taylor: T4(m) = 1 - m^2/2 + m^4/24
Chebyshev: C4(m) = (7/8) + (1/2) T2(m) + (1/8) T4(m), T_k = Chebyshev_k
Symbolic evaluations carry a:
x = (0.8, 0.3)
cos_Taylor*(x) = ( T4(0.8) , 0.3 ) ≈ ( 0.69627 , 0.3 )
cos_Chebyshev*(x)= ( C4(0.8) , 0.3 ) ≈ ( 0.69655 , 0.3 )
Magnitude errors reflect the classical approximants; alignment metadata is preserved for audit.
Implementation notes (manifest)
series_center = m0,radius = R,series_type = "Taylor" | "Chebyshev" | "Fourier-like"series_guard = "error" | "warn" | "sum_by_method:<name>"branchfor multi-valued targets;domain_real = <intervals>for principal real usageprecision_targets:(abs_tol, rel_tol)and remainder bound method- Alignment handling: strictly carry
a; useu = atanh(a)only in downstream multiplicative contexts (not needed for series evaluation itself) - Zero policy: if an approximant yields exact
0, record(0, +1)
Takeaway
Power series and analytic functions lift canonically via alignment carry: convergence, differentiation, integration, and error bounds are exactly those of the classical magnitude channel; the alignment channel is preserved for reproducibility. Uniform convergence ensures that collapse interchanges with limits safely, so classical series methods remain valid in symbolic space.
Navigation
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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