Manuscript VIII-ST@1 · VIII-ST.tex · section · fe979f3b

An exact algebraic finite catalogue

Source selection, affinity, and finite promises › An exact algebraic finite catalogue · lines 1358–1385

VIII-ST@1#eq:finite-table, VIII-ST@1#eq:algebraic

For an integer $q \geq 6$ , let

K_{q} = ⌊ \frac{2 ^{q}}{40} ⌋, T_{q} = {- K_{q}, \dots, K_{q}}, t_{m} = m 2^{- q}, μ_{m} = μ_{0} + 6 arctan t_{m} .

Every entry lies in $I$ , since

∣ μ_{m} - μ_{0} ∣ \leq 6 arctan (1/40) < 3/20 < π /20.

Its sector angle is

ϑ_{m} = \frac{π}{2} - 2 arctan t_{m} .

Consequently,

cos ϑ_{m} a_{m} = \frac{2 t _{m}}{1 + t _{m} ^{2}}, = \frac{2}{3} \frac{1 + t _{m}}{1 + t _{m} ^{2}}, sin ϑ_{m} b_{m} = \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}}, = \frac{3}{5} \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}} .

Displayed formulas in this passage

VIII-ST@1#eq:finite-table · line 1361
$K_{q} = ⌊ \frac{2 ^{q}}{40} ⌋, T_{q} = {- K_{q}, \dots, K_{q}}, t_{m} = m 2^{- q}, μ_{m} = μ_{0} + 6 arctan t_{m} .$
VIII-ST@1#eq:algebraic · line 1378
$cos ϑ_{m} a_{m} = \frac{2 t _{m}}{1 + t _{m} ^{2}}, = \frac{2}{3} \frac{1 + t _{m}}{1 + t _{m} ^{2}}, sin ϑ_{m} b_{m} = \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}}, = \frac{3}{5} \frac{1 - t _{m} ^{2}}{1 + t _{m} ^{2}} .$
unlabeled · line 1369
$∣ μ_{m} - μ_{0} ∣ \leq 6 arctan (1/40) < 3/20 < π /20.$
unlabeled · line 1374
$ϑ_{m} = \frac{π}{2} - 2 arctan t_{m} .$