Integral3D.ipynb

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   "source": [
    "using QuadGK"
   ]
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   "source": [
    "\\begin{align}\n",
    "\\iiint_{\\mathbb{R}^3}\n",
    "\\exp(-a r^2) ~\n",
    "\\mathrm{d}\\pmb{r}\n",
    "&=\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a x^2 -a y^2 -a z^2) ~\n",
    "\\mathrm{d}x\n",
    "\\mathrm{d}y\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a x^2) \n",
    "\\exp(-a y^2) \n",
    "\\exp(-a z^2) ~\n",
    "\\mathrm{d}x\n",
    "\\mathrm{d}y\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a y^2) \n",
    "\\exp(-a z^2)\n",
    "\\left[\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a x^2) \n",
    "\\mathrm{d}x\n",
    "\\right] ~\n",
    "\\mathrm{d}y\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\sqrt{\\frac{\\pi}{a}}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a z^2)\n",
    "\\left[\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a y^2) \n",
    "\\mathrm{d}y\n",
    "\\right]\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\sqrt{\\frac{\\pi}{a}}\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a z^2)\n",
    "\\left[\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a y^2) \n",
    "\\mathrm{d}y\n",
    "\\right]\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\sqrt{\\frac{\\pi}{a}}^2\n",
    "\\int_{-\\infty}^{\\infty}\n",
    "\\exp(-a z^2)\n",
    "\\mathrm{d}z \\\\\n",
    "&=\n",
    "\\sqrt{\\frac{\\pi}{a}}^3\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "dc350e2c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "evals  \tnumerical\n",
      "10^3\t1.001530395550591\n",
      "20^3\t0.9999875777006089\n",
      "50^3\t0.9999938466553702\n",
      "100^3\t1.0000001156363314\n",
      "200^3\t1.0000000000003697\n",
      "500^3\t1.0000000000003697\n",
      "1000^3\t1.0000000000003697\n",
      "∞\t1.0\n"
     ]
    }
   ],
   "source": [
    "a = π\n",
    "\n",
    "println(\"evals  \\tnumerical\")\n",
    "for n in [10, 20, 50, 100, 200, 500, 1000]\n",
    "    numerical  = (\n",
    "        quadgk(z ->\n",
    "        quadgk(y ->\n",
    "        quadgk(x ->\n",
    "        exp(-a*x^2 -a*y^2 -a*z^2)\n",
    "        , -Inf, Inf, maxevals=n)[1]\n",
    "        , -Inf, Inf, maxevals=n)[1]\n",
    "        , -Inf, Inf, maxevals=n)[1]\n",
    "    )\n",
    "    println(\"$n^3\\t\", numerical)\n",
    "end\n",
    "println(\"∞\\t\", sqrt(π/a)^3)"
   ]
  }
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