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DedekindEta at a complex τ at the working precision - #561
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…g precision (modular reduction, pentagonal sum with a tail bound); below the floor it stays unevaluated Fixes cortex-js#555
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Fixes #555
For Im τ ≥ 10^-3 above machine precision,
numerics/dedekind-eta-big.ts:η(τ+1) = e^{iπ/12}η(τ)andη(−1/τ) = √(−iτ)η(τ)(DLMF 23.15.1–2): the shift count is kept as an exact integer mod 24, so any Re τ is exact, and the √ factors as a complex product; the digits that an inversion at |τ| < 1 costs (1/|τ|² on the absolute error of τ) are counted and added to the working precision, up to a named cap;e^{iπτ/12} Σ (−1)^k q^{k(3k−1)/2}; its exponents are distinct, so the tail from exponent N is at most |q|^N/(1−|q|) and N is chosen from that bound (no convergence heuristic)._inexactNumericValue({re, im}), asapplyOnImaginaryAxisdoes), so nothing is padded or truncated to doubles.ETA_BIG_MIN_IMAGINARY = 1e-3. Below it the call stays unevaluated at these precisions; the reductions were measured only down to there. Other caps (named): 500 modular steps, 2,000 extra digits, 20,000 for the q exponent.evaluate()leaves exact operands exact.Accuracy against mpmath 1.3.0 (reference: the pentagonal sum evaluated directly, no reduction, at raised digits; it agrees with mpmath's
etato 5e-94 where that converges) on 110 cases at each precision (Re τ in 0, 0.3, −0.5, 0.25, 1/3, 12.7, 10^5 + 0.123456789, 10^20, −7.99999, 0.4999999999999, 0.001; Im τ in 0.001 to 100), error relative to |η|, in units of 10^−(p−1):No decline in the grid. The slowest case took 19 ms (100 digits).