r2_complex.py -- mpmath 1.3.0, 60 digits; 3 ellipses x 15 families

== ellipse a = 1, b = 0.6
  [PASS] N=3 tau=1 : lambda geom-Jacobi 1.6e-58, Lemma 3.1 5.61e-58, bridge 8.92e-57, Theorem B 5.34e-57, residues 7.36e-58, oddness 1.02e-55; gamma = -1.0; res F = (-6.2670747122e-64 - 0.475938875463j) = rho, res G/res F = gamma
  [PASS] N=4 tau=1 : lambda geom-Jacobi 8.17e-59, Lemma 3.1 1.74e-58, bridge 6.06e-58, Theorem A 7.73e-58, residues 6.99e-58, oddness 1.54e-56; c1 = 1.27058823529412, c2 = 0.0; res F = (-7.40654284169e-63 - 0.882352941176j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=5 tau=1 : lambda geom-Jacobi 5.49e-59, Lemma 3.1 3.62e-59, bridge 4.67e-58, Theorem B 2.67e-58, residues 9.27e-59, oddness 3.02e-57; gamma = 0.630882535378537; res F = (5.29852680213e-63 - 0.392943873639j) = rho, res G/res F = gamma
  [PASS] N=5 tau=2 : lambda geom-Jacobi 1.97e-59, Lemma 3.1 8.54e-57, bridge 6.97e-55, Theorem B 1.29e-55, residues 1.05e-56, oddness 1.69e-54; gamma = -1.21648419678105; res F = (-3.39941325298e-63 - 0.479949455789j) = rho, res G/res F = gamma
  [PASS] N=6 tau=1 : lambda geom-Jacobi 2.13e-58, Lemma 3.1 6.55e-58, bridge 2.24e-57, Theorem A 3.16e-57, residues 6.77e-57, oddness 3.86e-56; c1 = 2.32837125932538, c2 = 2.32837125932538; res F = (-1.08249472302e-62 - 0.695268608165j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=7 tau=1 : lambda geom-Jacobi 2.03e-58, Lemma 3.1 7.47e-58, bridge 2.7e-57, Theorem B 5.47e-57, residues 1.16e-57, oddness 6.21e-56; gamma = 1.23681541479136; res F = (-1.32842992763e-62 - 0.309017422108j) = rho, res G/res F = gamma
  [PASS] N=7 tau=2 : lambda geom-Jacobi 8.91e-59, Lemma 3.1 4.53e-58, bridge 2.3e-57, Theorem B 2.98e-57, residues 1.56e-57, oddness 4.68e-56; gamma = -0.478887026288916; res F = (2.35774947127e-62 - 0.462066945471j) = rho, res G/res F = gamma
  [PASS] N=7 tau=3 : lambda geom-Jacobi 3.38e-59, Lemma 3.1 8.55e-56, bridge 8.79e-52, Theorem B 1.81e-54, residues 1.48e-55, oddness 2.37e-53; gamma = -1.21947473509684; res F = (4.0261207242e-63 - 0.479999375572j) = rho, res G/res F = gamma
  [PASS] N=8 tau=1 : lambda geom-Jacobi 2.19e-58, Lemma 3.1 1.13e-57, bridge 3.99e-57, Theorem A 8.65e-57, residues 3.45e-56, oddness 6.61e-56; c1 = 2.81697877150571, c2 = 3.94046306433779; res F = (3.10789932319e-62 - 0.553762715046j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=8 tau=3 : lambda geom-Jacobi 4.31e-59, Lemma 3.1 3.94e-57, bridge 1.02e-55, Theorem A 3.04e-56, residues 8.08e-56, oddness 4.7e-55; c1 = 1.87174374866732, c2 = -2.23231188902984; res F = (3.44166852945e-62 - 0.959092804727j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=9 tau=2 : lambda geom-Jacobi 3.1e-58, Lemma 3.1 1.21e-57, bridge 5.74e-57, Theorem B 5.76e-57, residues 5.46e-57, oddness 1.14e-55; gamma = 0.361039135450947; res F = (9.58102785547e-63 - 0.417275910463j) = rho, res G/res F = gamma
  [PASS] N=9 tau=4 : lambda geom-Jacobi 1.99e-58, Lemma 3.1 1.13e-52, bridge 3.75e-48, Theorem B 3.07e-51, residues 2.51e-52, oddness 4.02e-50; gamma = -1.21951173263729; res F = (-5.69734064746e-64 - 0.479999992291j) = rho, res G/res F = gamma
  [PASS] N=10 tau=3 : lambda geom-Jacobi 1.66e-58, Lemma 3.1 1.24e-57, bridge 8.56e-57, Theorem A 4.98e-57, residues 4.95e-56, oddness 9.88e-56; c1 = 6.56213785196372, c2 = -4.32860523967129; res F = (-3.23608948775e-62 - 0.935388806852j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=11 tau=4 : lambda geom-Jacobi 2.07e-58, Lemma 3.1 9.44e-57, bridge 2.45e-55, Theorem B 6.77e-56, residues 9.33e-57, oddness 1.46e-54; gamma = -1.16445178051266; res F = (1.06350358753e-63 - 0.47905834917j) = rho, res G/res F = gamma
  [PASS] N=12 tau=5 : lambda geom-Jacobi 3.09e-58, Lemma 3.1 8.57e-55, bridge 5.1e-52, Theorem A 1.01e-53, residues 2.81e-53, oddness 1.53e-52; c1 = 1.88203397418342, c2 = -2.29452902203683; res F = (2.84867032373e-62 - 0.959988761367j) = 2 rho; F, G regular and zero at iK'-D

== ellipse a = 3, b = 2
  [PASS] N=3 tau=1 : lambda geom-Jacobi 2.85e-58, Lemma 3.1 8.8e-58, bridge 9.48e-57, Theorem B 7.57e-57, residues 1.29e-57, oddness 1.42e-55; gamma = -1.0; res F = (-1.03311777074e-61 - 4.44097122862j) = rho, res G/res F = gamma
  [PASS] N=4 tau=1 : lambda geom-Jacobi 1.19e-58, Lemma 3.1 3.08e-58, bridge 1.07e-57, Theorem A 1.41e-57, residues 1.34e-57, oddness 2.52e-56; c1 = 132.923076923077, c2 = 0.0; res F = (-8.09022371939e-62 - 8.30769230769j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=5 tau=1 : lambda geom-Jacobi 1.72e-58, Lemma 3.1 3.86e-58, bridge 1.44e-57, Theorem B 3.46e-57, residues 1.01e-57, oddness 3.05e-56; gamma = 0.626199152618636; res F = (-1.19872047222e-61 - 3.71932213547j) = rho, res G/res F = gamma
  [PASS] N=5 tau=2 : lambda geom-Jacobi 2.74e-58, Lemma 3.1 2.42e-56, bridge 2.42e-54, Theorem B 3.15e-55, residues 3.46e-56, oddness 4.21e-54; gamma = -1.27870845135847; res F = (9.73865428005e-62 - 4.4715438309j) = rho, res G/res F = gamma
  [PASS] N=6 tau=1 : lambda geom-Jacobi 1.41e-58, Lemma 3.1 3.47e-58, bridge 1.4e-57, Theorem A 2.07e-57, residues 3.8e-57, oddness 1.93e-56; c1 = 236.504898586393, c2 = 236.504898586393; res F = (-1.33887505215e-61 - 6.59890900074j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=7 tau=1 : lambda geom-Jacobi 1.61e-58, Lemma 3.1 7.38e-58, bridge 2.47e-57, Theorem B 7.68e-57, residues 1.16e-57, oddness 5.87e-56; gamma = 1.24050584662175; res F = (2.62077669783e-62 - 2.93752450857j) = rho, res G/res F = gamma
  [PASS] N=7 tau=2 : lambda geom-Jacobi 6.74e-59, Lemma 3.1 2.8e-58, bridge 1.51e-57, Theorem B 1.81e-57, residues 9.79e-58, oddness 2.66e-56; gamma = -0.466964050044426; res F = (-4.34517180046e-62 - 4.33188621231j) = rho, res G/res F = gamma
  [PASS] N=7 tau=3 : lambda geom-Jacobi 1.29e-59, Lemma 3.1 6.9e-56, bridge 3.69e-53, Theorem B 1.25e-54, residues 1.38e-55, oddness 1.68e-53; gamma = -1.28556440154368; res F = (3.90457745706e-62 - 4.47212336899j) = rho, res G/res F = gamma
  [PASS] N=8 tau=1 : lambda geom-Jacobi 4.92e-59, Lemma 3.1 1.22e-58, bridge 6.9e-58, Theorem A 1.26e-57, residues 3.94e-57, oddness 6.78e-57; c1 = 283.996222542578, c2 = 398.849339825951; res F = (7.81295314121e-62 - 5.26922192453j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=8 tau=3 : lambda geom-Jacobi 1.94e-58, Lemma 3.1 4.78e-57, bridge 1.77e-55, Theorem A 3.29e-56, residues 1.15e-55, oddness 5.03e-55; c1 = 210.595088047821, c2 = -260.891670204362; res F = (2.01154107126e-61 - 8.93604362498j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=9 tau=2 : lambda geom-Jacobi 6.95e-59, Lemma 3.1 3.76e-58, bridge 1.28e-57, Theorem B 2.03e-57, residues 1.71e-57, oddness 3.37e-56; gamma = 0.356020323364332; res F = (-3.47917602205e-62 - 3.94101185918j) = rho, res G/res F = gamma
  [PASS] N=9 tau=4 : lambda geom-Jacobi 7.25e-59, Lemma 3.1 8.29e-54, bridge 6.63e-50, Theorem B 1.94e-52, residues 2.14e-53, oddness 2.6e-51; gamma = -1.28571109478157; res F = (-4.46671506761e-62 - 4.47213568709j) = rho, res G/res F = gamma
  [PASS] N=10 tau=3 : lambda geom-Jacobi 2.39e-58, Lemma 3.1 1.68e-57, bridge 1.01e-56, Theorem A 6.28e-57, residues 7.41e-56, oddness 1.22e-55; c1 = 706.077720939541, c2 = -455.713183575409; res F = (1.67349885951e-61 - 8.75486173192j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=11 tau=4 : lambda geom-Jacobi 7.03e-59, Lemma 3.1 1.75e-57, bridge 3.57e-56, Theorem B 1.03e-56, residues 1.99e-57, oddness 2.43e-55; gamma = -1.19856508501827; res F = (-1.08021578676e-61 - 4.4641959603j) = rho, res G/res F = gamma
  [PASS] N=12 tau=5 : lambda geom-Jacobi 2.13e-58, Lemma 3.1 1.82e-55, bridge 5.9e-53, Theorem A 1.91e-54, residues 7.19e-54, oddness 2.85e-53; c1 = 212.993451364006, c2 = -273.630072880767; res F = (-2.2910905857e-61 - 8.94409936313j) = 2 rho; F, G regular and zero at iK'-D

== ellipse a = 2.5, b = 0.55
  [PASS] N=3 tau=1 : lambda geom-Jacobi 1.39e-58, Lemma 3.1 1.17e-56, bridge 1.27e-54, Theorem B 2.21e-55, residues 4.15e-57, oddness 4.13e-54; gamma = -1.0; res F = (1.25151582889e-62 - 1.3409306306j) = rho, res G/res F = gamma
  [PASS] N=4 tau=1 : lambda geom-Jacobi 1.25e-58, Lemma 3.1 8.34e-58, bridge 4.66e-57, Theorem A 6.02e-57, residues 1.7e-57, oddness 1.32e-55; c1 = 3.17388401373522, c2 = 0.0; res F = (3.9140730248e-62 - 2.62304463945j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=5 tau=1 : lambda geom-Jacobi 7.91e-59, Lemma 3.1 2.8e-58, bridge 8.12e-58, Theorem B 2.7e-57, residues 5.64e-58, oddness 3.82e-56; gamma = 0.713145802151012; res F = (-2.84487209663e-62 - 1.22132848684j) = rho, res G/res F = gamma
  [PASS] N=5 tau=2 : lambda geom-Jacobi 2.12e-58, Lemma 3.1 1.28e-52, bridge 1.06e-48, Theorem B 4.05e-51, residues 3.49e-53, oddness 5.04e-50; gamma = -1.0247963016002; res F = (3.22849303356e-64 - 1.34131226506j) = rho, res G/res F = gamma
  [PASS] N=6 tau=1 : lambda geom-Jacobi 1.85e-58, Lemma 3.1 4.36e-58, bridge 2.11e-57, Theorem A 2.27e-57, residues 2.8e-57, oddness 4.08e-56; c1 = 9.15188607324622, c2 = 9.15188607324622; res F = (9.60951455871e-63 - 2.21714592851j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=7 tau=1 : lambda geom-Jacobi 1.91e-58, Lemma 3.1 5.63e-58, bridge 1.95e-57, Theorem B 3.98e-57, residues 6.85e-58, oddness 6.92e-56; gamma = 1.1745392842954; res F = (-1.47371211414e-62 - 1.00028858275j) = rho, res G/res F = gamma
  [PASS] N=7 tau=2 : lambda geom-Jacobi 2.97e-58, Lemma 3.1 7.28e-57, bridge 9.85e-56, Theorem B 9.69e-56, residues 1.9e-56, oddness 1.36e-54; gamma = -0.643703473854352; res F = (-1.21543267146e-62 - 1.33425244932j) = rho, res G/res F = gamma
  [PASS] N=7 tau=3 : lambda geom-Jacobi 3.07e-58, Lemma 3.1 7.37e-49, bridge 8.16e-43, Theorem B 3.26e-47, residues 2.81e-49, oddness 4.06e-46; gamma = -1.02480016337396; res F = (1.1242752211e-62 - 1.3413123238j) = rho, res G/res F = gamma
  [PASS] N=8 tau=1 : lambda geom-Jacobi 1.89e-58, Lemma 3.1 5.35e-58, bridge 2.33e-57, Theorem A 3.18e-57, residues 9.62e-57, oddness 4.68e-56; c1 = 12.6451308116094, c2 = 16.482217333963; res F = (2.40332819645e-62 - 1.80931341005j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=8 tau=3 : lambda geom-Jacobi 2.84e-58, Lemma 3.1 4.11e-54, bridge 2.0e-51, Theorem A 6.03e-53, residues 3.44e-53, oddness 9.72e-52; c1 = 3.36772530658539, c2 = -3.45019679640764; res F = (-5.57579738031e-62 - 2.68261517329j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=9 tau=2 : lambda geom-Jacobi 8.87e-59, Lemma 3.1 7.43e-58, bridge 3.74e-57, Theorem B 5.93e-57, residues 3.01e-57, oddness 1.12e-55; gamma = 0.451617168025606; res F = (6.73805487239e-62 - 1.27199314616j) = rho, res G/res F = gamma
  [PASS] N=9 tau=4 : lambda geom-Jacobi 7.05e-59, Lemma 3.1 6.64e-46, bridge 9.35e-38, Theorem B 3.78e-44, residues 3.25e-46, oddness 4.7e-43; gamma = -1.02480016396793; res F = (2.24285310155e-62 - 1.34131232381j) = rho, res G/res F = gamma
  [PASS] N=10 tau=3 : lambda geom-Jacobi 1.14e-58, Lemma 3.1 6.88e-57, bridge 1.28e-55, Theorem A 5.85e-56, residues 1.48e-55, oddness 1.02e-54; c1 = 14.6325870291354, c2 = -12.0261501491936; res F = (6.86529548018e-62 - 2.67580017654j) = 2 rho; F, G regular and zero at iK'-D
  [PASS] N=11 tau=4 : lambda geom-Jacobi 1.61e-58, Lemma 3.1 9.75e-55, bridge 2.61e-52, Theorem B 1.96e-53, residues 2.02e-55, oddness 2.86e-52; gamma = -1.02345563779623; res F = (-1.23442380695e-62 - 1.34129185913j) = rho, res G/res F = gamma
  [PASS] N=12 tau=5 : lambda geom-Jacobi 2.47e-58, Lemma 3.1 1.38e-50, bridge 9.13e-46, Theorem A 3.03e-49, residues 1.74e-49, oddness 4.88e-48; c1 = 3.36775677988867, c2 = -3.45127753891495; res F = (-7.33627564037e-62 - 2.68262464616j) = 2 rho; F, G regular and zero at iK'-D

summary (a)-(e): worst |lambda_geom - lambda_Jacobi|/b^2 3.1e-58, Lemma 3.1 6.64e-46, bridge geometry = Jacobi (Prop. 3.4, Lemma 4.3) 9.35e-38, Theorems A/B at complex phases 3.78e-44, residues 3.25e-46, oddness 4.7e-43

(f) Lemma 3.3(a) at 250 digits
  [PASS] a=1 b=0.6: K - Delta -> arsinh(c/b) (error proportional to 1 - lambda/b^2), K -> infinity, Delta/K -> 1; sn(K - Delta) = a''/a : at 1 - lambda/b^2 = 1e-10, 1e-20, 1e-40, 1e-80: K = 13.187, 24.7, 47.726, 93.777; K - Delta - arsinh(c/b) = 4.67e-11, 4.67e-21, 4.67e-41, 4.67e-81; 1 - Delta/K = 0.0833, 0.0445, 0.023, 0.0117; identity 1.44e-172
  [PASS] a=3 b=2: K - Delta -> arsinh(c/b) (error proportional to 1 - lambda/b^2), K -> infinity, Delta/K -> 1; sn(K - Delta) = a''/a : at 1 - lambda/b^2 = 1e-10, 1e-20, 1e-40, 1e-80: K = 13.011, 24.524, 47.55, 93.601; K - Delta - arsinh(c/b) = 5.28e-11, 5.28e-21, 5.28e-41, 5.28e-81; 1 - Delta/K = 0.074, 0.0392, 0.0202, 0.0103; identity 2.66e-172
  [PASS] a=2.5 b=0.55: K - Delta -> arsinh(c/b) (error proportional to 1 - lambda/b^2), K -> infinity, Delta/K -> 1; sn(K - Delta) = a''/a : at 1 - lambda/b^2 = 1e-10, 1e-20, 1e-40, 1e-80: K = 14.389, 25.901, 48.927, 94.979; K - Delta - arsinh(c/b) = 2.84e-11, 2.84e-21, 2.84e-41, 2.84e-81; 1 - Delta/K = 0.153, 0.0847, 0.0449, 0.0231; identity 1.02e-171
  [PASS] a=1 b=0.01: K - Delta -> arsinh(c/b) (error proportional to 1 - lambda/b^2), K -> infinity, Delta/K -> 1; sn(K - Delta) = a''/a : at 1 - lambda/b^2 = 1e-10, 1e-20, 1e-40, 1e-80: K = 17.504, 29.017, 52.043, 98.095; K - Delta - arsinh(c/b) = 2.5e-11, 2.5e-21, 2.5e-41, 2.5e-81; 1 - Delta/K = 0.303, 0.183, 0.102, 0.054; identity 1.96e-172

(g) Lemma 6.1
  [PASS] 2 + gamma = 2q/(a^4b^4 - c^4 lambda^2): q > 0 and denominator > 0 on [0, b^2], b/a from 0.001 to 0.999 (2001 points each) : min q/(a^4b^4) = 9.99999e-7, min den/(a^4b^4) = 2.0e-6
  [PASS] odd N = 3..41, every tau, five moduli k^2 = 0.05 .. 1 - 1e-9 (890 families, 30 digits; ellipse from the caustic by a = a'' dn D/cn D, b = b''/cn D): gamma(sn-dn form of Theorem B) = gamma(lambda-form of Lemma 6.1), 2 + gamma > 0 : largest difference 2.24e-27, smallest 2 + gamma 0.00586983, confocality/half-step check 5.38e-22

TOTAL checks: 51, failures: 0
running time: 102 s
