{
  "complex_polynomial_root": "P(z)=(z-i)^2",
  "distance": "sqrt(1+xi^2)",
  "strength": {
    "A": "d^2+2",
    "B": "2*sqrt(d^2+1)",
    "ratio": "(d^2+2)/(2*sqrt(d^2+1))",
    "lower_bounds": [
      "d/2",
      "(1+d)/4"
    ],
    "ordinary_derivatives_not_factorial_normalized": true
  },
  "real_collision": {
    "P": "z^2",
    "A": "xi^2+2",
    "B": "2*sqrt(1+xi^2)",
    "ratio_at_zero": 1,
    "raw_ratio": "|xi|/2 only for xi!=0",
    "raw_at_zero": "undefined; plotted open circle"
  },
  "heat": {
    "P": "z1^2+i*z2",
    "operator": "partial_x2-partial_x1^2",
    "actual_complex_curve": "((1+i)*s,-2*s^2)",
    "ambient_real_dimension": 4,
    "real_coordinates": "(s,-2*s^2)",
    "imaginary_coordinates": "(s,0)",
    "imaginary_norm": "s for s>0",
    "magnitudes": [
      "sqrt(2)*s",
      "2*s^2"
    ],
    "plots_are_exact_scalar_magnitudes": true,
    "solution": "integral_0^infty exp(-a0*s+(i-1)*x1*s-2*i*x2*s^2) ds",
    "parameter_measure": "real Lebesgue ds, not full complex-surface area",
    "domain": "x1>-a0, x2 real",
    "a0": 1,
    "spatial_D_derivative": "(1+i)^j*j!/a0^(j+1)",
    "time_D_derivative": "(-2)^j*(2*j)!/a0^(2*j+1)",
    "convention": "D=-i*partial",
    "plotted_orders": [
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
      12,
      13,
      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23,
      24
    ]
  },
  "exact_proof_locators": [
    "EB1-EB5",
    "EB9-EB16",
    "L154.1-L154.10",
    "L154.13"
  ],
  "human_source": "Lars Hormander, Analysis of Linear PDE II, two exercises on printed page291; Theorems11.1.7,11.4.1,11.4.8",
  "original_sources": true,
  "blender_useful": false,
  "blender_reason": "Exact scalar strengths, four-dimensional curve coordinates and derivative orders are fully explained by scientific plots."
}