14_Fourier_Analysis_I.ipynb

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    "# Lecture 14: Fourier Analysis I\n",
    "\n",
    "## The Dirac Delta function as a limit\n",
    "\n",
    "Consider the function\n",
    "$$\n",
    "\\hat{E}_{\\alpha} (k) = \\frac{\\alpha}{\\pi (\\alpha^2 + k^2)}.\n",
    "$$\n",
    "\n",
    "In the limit $\\alpha\\to 0$, this function becomes taller and slimmer, approaching\n",
    "a peak of infinite height and zero width: A Dirac $\\delta$ \"function\" (actually a distribution).\n",
    "\n",
    "Note: This is not the only limit expression for the $\\delta$ \"function\". There are infinitely many."
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     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "E(α, k) = α/(π*(α^2 + k^2))\n",
    "\n",
    "fig, ax = subplots()\n",
    "\n",
    "xs = LinRange(-1, 1, 1001)\n",
    "\n",
    "@manipulate for α=slider(10.0.^LinRange(-2, 0, 20), value=1.0, label=\"α\")\n",
    "    withfig(fig, clear=false) do\n",
    "        ax.cla()\n",
    "        E_α(x) = E(α, x)\n",
    "        ax.plot(xs, E_α.(xs), label=\"α = $(round(100α)/100)\")\n",
    "\n",
    "        ax.set_xlim(-1, 1)\n",
    "        ax.set_ylim(0, 10)\n",
    "\n",
    "        ax.set_xlabel(L\"$k$\")\n",
    "        ax.set_ylabel(L\"E_{\\alpha}(k)\")\n",
    "        ax.set_title(L\"Delta function $\\delta(k)$ as a limit\")\n",
    "\n",
    "        ax.legend()\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_prime(α, k) = -2k*α/(π*(α^2 + k^2)^2)\n",
    "\n",
    "fig, ax = subplots()\n",
    "\n",
    "xs = LinRange(-1, 1, 1001)\n",
    "\n",
    "@manipulate for α=slider(10.0.^LinRange(-2, 0, 20), value=1.0, label=\"α\")\n",
    "    withfig(fig, clear=false) do\n",
    "        ax.cla()\n",
    "        E_α(x) = E_prime(α, x)\n",
    "        ax.plot(xs, E_α.(xs), label=\"α = $(round(100α)/100)\")\n",
    "\n",
    "        ax.set_xlim(-1, 1)\n",
    "        ax.set_ylim(-50, 50)\n",
    "\n",
    "        ax.set_xlabel(L\"$k$\")\n",
    "        ax.set_ylabel(L\"E_{\\alpha}(k)\")\n",
    "        ax.set_title(L\"Delta function derivative $\\delta'(k)$ as a limit\")\n",
    "\n",
    "        ax.legend()\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Gibbs phenomenon\n",
    "We consider the Fourier series of the periodic, discontinuous sawtooth function\n",
    "\n",
    "$$\n",
    "f(t) = t \\qquad 0 < t \\leq 1,\n",
    "$$\n",
    "\n",
    "continued periodically with period $T=1$.\n",
    "The Fourier series is\n",
    "\n",
    "$$\n",
    "f(t) = \\frac{1}{2} - \\frac{1}{\\pi} \\sum_{\\nu=1}^\\infty \\frac{1}{\\nu} \\sin(2\\pi \\nu t).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BpgEDHU3oamss+TdjZ0m+TgCA6eUkZqPrt8LSu6WTL2g4M7m2MRxVeEpZq8oKUIVnNWtpnUSGEh4OkIZHZz3BMkA6CyvrHzs4/TRiw2K1XissaVN0Ls7GkMjk0Rqsx11dbbf8u2HQ6PE4AUA8ncOF2aImpYzVAZlZllLOJFV0KOHhAO0uddbys2CQB4jJZsTdXn+fGBsdZwEUM8aIpcpaV67K6qc4McYnIyhoQF+4Ed2hxlv+nSo8JnPRFKaWk/D7gKEBSng8De0ugcWVNK4VW2GHNxgQXp9sNU3D6MTGi7jX48QwE+m13yvSh+lstAsnDY+JIYRfY4NmVsMoWGyDtqsnhJag+IaDDEp4OEAaHv0STAC4s6v1llZYBu2YdK4uxrFcbIXds6oVFqA4WUnn8jg3pR87lDMcZFALsU45524rVAkzoVhtnrXMGUWHEh4O0AK1/j1HDLojSoftwsu1wgKAv/gpr8cJAM5NR5HJb9wKS1VW4GY8g6uLcQD6kVY5yONJRzcc3KAaRvOVgWwOywxKeDhAu0urT8o6u3A/aXgA0jpVgjnRrt8KS7tx03BwZ2cLOloCZb+GPJ50ri8lcCOeQaDOj319t1ZZAdrIMjK5Al6bKl7rQhUewufx7qONDAcZbLny+mRrXcTLQXEyWesSzNWQhmdzmw5r1cLLCTWL1f7+EIL1dWW/hqQKOudnokjnCuhobsDtnS1uP05FUMLDAa93aV2c01th24L1uKvrVsNBBt1NA6ykc7g0xwwH1xJLUpwY63WzWaEq6+bE3dbrXTwcqnXNGRlUadWxJtKyGA4yKOHhgHllgssP4hJM0Da0huEggzQ8wPjEMgoa0N/eVLYVFqDFmzEbSWE6ktpUK6zXPZ7yBc1oHNiMuBvwbqwA02F5fasD/aPHh6Exvx9cQxcmMpTwcMDrFZ7RTQiWAdJZAJu7bZiMLHVMw8GNW2G9rk25PB9DPJNHS6AOu3puNRxkWHfoXo1VIpPD+Rm9yrq+1QFVWgHL/L6G5lBkKOHhgNfv8TF0FhsMCPK12JwmhbQDOps9zgKsuieODyQw7IhmaA3DQYb1RMKr79f4ZAT5goaeUCN6w01rfh1p6YD5WAqTN5Pw+fR3SzYo4eGA2erpvYGxFM/gzWIr7EYlT6/vmEoNBzeu8Hg1ToyNriqx4vUq62YqhwBpeACLc/c61R2ANDyAmUjv6m5Dq0SGgwxKeDjg5YV8rHgWvnNbC9qby7fCMrycGALAWzcSWIpnEKj3Y19f+UswAbrDByhthd1MKd3vcYOZjUz0GCUaHo/Gar177KwYfli8H0hgzNvk5TvOAijh4QIbGF7cXa532/BqvF65YO3oaxkOMrweJwB4fSaKTLEVdsc6hoMML2t4lhMZvLFQrLJumPB4W8OjGw5ubhE3m1G8FyfG6LWNK9IiQwkPB0wNj/cGxmZ3S4D1Hh/vxQmw7JY2PPrTP3o1ToCZHG62FdbLegt2TLpjazO2rGE4WA4vxmryZhKLKxk01PnWNBxkeF0Xls0XMD7Fjv+owkMU8Wr7Yr6g4czE+vbsVoybmr3avn9tc5MHVXg2d1WJFS/HbDO3yTNKKzzcHklY2Hu1ry+MxobyhoMMr3eVXpiJIZUtINzUgNu3ymU4yKCEhwNenWwvzemtsK3BetzdvXYrLMPL2pR4OocLs/olmBstTFTh2bzDMsOrmw7A0s22Ga2TtVjmwVhVcifUOs1unsAqhF/PX01kKOHhgFfbiNmAGBoMr9sKy/BqYgjorbAFDegNN6InXN5wkOHlOAHAXDSFqeWkbji4yVZYr3bUFCyGg5uphnldwzNaSUXa4xWeSuQKokIJDwe8Kpg07NnXcXa14mWdRTVaJ8B7CzhgVizurqQV1qObjisLK4ilc2gO1GFXBVVWwHuxSmXzeH26WGXdjCbF4/ezbXSbvAxQwsMBo7jhsYFh2LNv4GfB8LKDcCUmeqSzqOw4C/BuVYwd0RwYCKO+buPp3efhd2t8MoJcQUN3KIi+DaqsgLcrPIsraVxfSsDnA4YlNBxkUMLDAS9OtsuJDK6yVtjNVng8qk3RW2E3v4iXmsN5K1bA5q8qseJVG55qjh286odlbDoGN9f552VdGEuk7+5qQ1tjg8tPUz2U8HDAi0dabAG/vbMFHZtshfViYggA15cSuBHPIFDnx/7+9VthAcBnGaVei1UmV8D45OYNBxle1fBUKu4GvFtp3cxt8la8+k4BlWmdRIYSHg54UZtSyRENw+vi7n39IQTr12+FBUo1PF6L1fmZKNK5ylthvfhuRZJZXJ5fAUDjcCM0TavoqhLAGideTyUulXSziQwlPBzwogV5pZMH4N2dpbEL3+TRn5fvOxqtshXWdMXl8lhCMlbchd+2tRmdrcFNf58Xr8KZvJnEQiyNer8P+/vXvtbFihcr9wCQy1urrFThIVbhtdJnvqAZk20lO0uvageqLaUDXoxV5Yk04M13y9SkVLYo+T2opWNHNPv6QhsaDjK8quG5MBtDMptHqLEeOztb3X6cmqCEhwNe211emV/BSgWtsAyvxQkAEpkczs/EAFReSge8tQsHqvf+8KI+zEgOK7T992Kl1XpVyWbx2kaWwRLp4e0d0hoOMoRJeJ5++mns2LEDjY2NOHToEF555ZV1v/6pp57Crl270NTUhMHBQXz6059GKpVy6GnXx2saHsNwcKB9U62wDC9qB16bjCBfbIXt3UQrLOBdc7j5WAqTN5Pw+XQzy0rw2j1thYK26RvSV+O1+QqoTnPoxSQasFZZ5T7OAgRJeF544QUcP34cJ06cwMjICIaGhnD06FHMz8+X/frvfve7+OxnP4sTJ07g/Pnz+Pa3v40XXngBf/Znf+bwk5fHay2xhqCtwvNdL04g1iOazbTCAqUVHs1D1TCmdaqmFdZrupQ3FlYQS+XQ2ODHrp7NV1kB743DVDaPc9Obu9alHF5KDAHrlRJyC5YBQRKeJ598Eo8++igeeeQR7N27F8888wyam5vx3HPPlf36n//853j729+OD33oQ9ixYwfe/e5344Mf/OCGVSGn8Frp02hZ3KQIl2FWR70RJ6BanxRvangq1TpZ8ZouhSWHBwba0VBBlRXwnh/W2SndcHBbWxADHU2b/j5jXuf1YAJyYyWNazcSAOQ2HGS4nvBkMhm8+uqrOHLkiPE5v9+PI0eO4PTp02W/521vexteffVVI8G5evUqfvSjH+F973vfmr8nnU4jGo2W/OGFl3ZMkUQWV6pohQW8twsvNRysXNwNeCdWQOXdbFa8NAaB2u45YroMr8Xq4GD7pqusgDkOvVThYWPwrq5WhJvkNRxkbPJiGn4sLi4in8+ju7u75PPd3d24cOFC2e/50Ic+hMXFRfz6r/86NE1DLpfDxz/+8XWPtE6ePIkvfvGLtj77mnhoYLDrJHZsbcbWClphAe9pByZvJrG4kkZD3eZbYYHV9v/eiFU2X8D4JBPhVr6z9FrVYqQKTQrDa3qn0WrF3X4PirtreK9ExPUKTzW8/PLLeOKJJ/CNb3wDIyMj+Jd/+Re8+OKL+NKXvrTm9zz22GOIRCLGn4mJCW7P56X2xWqcXRle3YXv7QtvuhWW4TWB94WZGFLZQtWtsF56t6Ip03CwqgqPh2KlGw7WKO72QqCKjFZpCyEqrld4Ojs7UVdXh7m5uZLPz83Noaenp+z3fOELX8BHPvIRfOxjHwMA3HPPPYjH4/i93/s9fO5zn4Pff2seFwwGEQxWVoGoFi9dMjdSxT1HDMOg0QNxAqxupdVoUnzIa5pnBAQjNbbCGhUeDwTszMQyNA0Y3NKEbW2Vz3E+Q5uifqymIynMRXXDwXsqqLIC1jh5g1y+gDOT1VXDRMX1Ck8gEMB9992HU6dOGZ8rFAo4deoUDh8+XPZ7EonELUlNXZ2+YxZh8fSKr0WhxHCw+p2l6nFijNYUK/2jVzaX1VwYasXYdHigq23kWvVaJ8DybnkgVuy92tMbQlOgsiqr1zQ8F+diSGTyaAvW485tchsOMlyv8ADA8ePH8dGPfhT3338/HnjgATz11FOIx+N45JFHAAAPP/ww+vv7cfLkSQDAgw8+iCeffBIHDx7EoUOHcOXKFXzhC1/Agw8+aCQ+buKV44crxVbYpoY67K6wFdaK6nEC9FbY141W2Go0KT4AmidiBVTvsMzwyhgETB1dtcmhl2LFksNqq6yAhzZoxTE4XOG1LiIjRMLz0EMPYWFhAY8//jhmZ2cxPDyMl156yRAyX79+vaSi8/nPfx4+nw+f//znMTU1hW3btuHBBx/El7/8Zbf+E0rweWRgsN3SgYFwRYaDDC8d/Y1P6q2wXW1B9LdvvhWW4SWB9+JKGteX9FbYoSpbYb2yOOmGg7UdO3glVoDV6qDyWHkpMQTU8t9hCJHwAMCxY8dw7Nixsv/28ssvl/y9vr4eJ06cwIkTJxx4ssrxSunT2C3RRLshVmfXSlphGd6KVe2tsF7R8FxdjCOSzCJY78ee3lBVP8PvEQ1POmdWWas5/vPKRpYxqpDDMsN1DY+KeGVxGqnyskKGl7rZavFJAShWleKVziNrlbVSw0GGzyP6sLNTUWTyBXS2BjC4pfIqq1c2sgBwM57Bm4txANVrw0SEEh4OeOH4IZK0tMJWWeHxSolYb4W1pxqmeqwA68WO1e8svTAGgdq1ToB3xuGo5Yimliqr6okhYOrC7tjWgnCz/IaDDEp4OOCF9sUzxY6j7Vua0Vmh4SDD55FFfGo5iYVYda2wBh5ZlHL5AsYnIwBqa4X1ikncqA06C69chVOriZ6XDBpNcbc61R2AEh4ueKH0aYcDp1fupmG78L19oYoNBxleidWF2RiS2TzaGmtrhfXCXVor6RwuzsUA1Kaz8MoRfK0mel5JogGzwqOSYBmghIcLXih92uHA6RVvGdNwsPZYqbyAA2bFYniwtlZYL9zTxgwH+9ub0BVqrPrneEHDMxNJYiaSQp3fhwMD1VVZvXL0ly9oGDOO4NURLAOU8HBB9Xt89FZY+4SlqsaJYRoO1r4LV3lRAmq7qsSKFzQ8RiJdowuuF/Rh7Ihmd08bmgPVNSd7ZQxemoshnsmjNViPu7qq91cTEUp4OKB6ifjq4gqiqRwaG/zY3VvDgPDAjkk3HCxqUkhYuiG1XFViRfUxCFgS6Sq7JBleSA7t2KB5RcPDxuDQYBh1ihgOMijh4YDqixPTpBzob6+6FRbwxqJ0diqCbF5DZ2sQAx2Vt8IyjCMahe3/b6yk8dYN3XCw1lZY1auHmmapstpU4VFZIGYaDtpRZVU4UFDvwlArlPBwQPXSp9EZUuP5rhc0POYRTXWGgwxDw6PwqsTuZbOjFVZ1Xcqbi3HcTGQRqPdjb5WGgwzVY5XO5XF2qnrDQYYH8kIA9vhgiQolPBxQXcNjV8ui6rtwwL7JwwvVMDsnWq9UWe/pDyNQX9s0rnrl4vVp3XBwS0sAt21trvrnqL6RBYDlRAZXF3TDweEaj0pFhBIeDqi8OMVSWVya11thaxHhAuprB3TDQXs0KarHCqj9qhIrqi9Otd4mb8UryeG9NVZZVY8TYOrCdna2oKMl4PLT2A8lPBxQeWCcmYhA04CBjiZ0tVXfCguofzfNTCSFuWgadX4f7qmyFZahept1Ll/Amcnau9kYhi5F0QMIOxyWGap7PNl1CaYXKtKj19T032FQwsMBlXeXdh47qG7QyGK1p7f6VliGvzhSVZ1sL82tIGFjK6zK+rCVdA4XZ4uaFBvHoarv1th1exJpL9xnN2JTrESFEh4OqHz8YIfDMkN151I77dlVTqIB870aHmy3pRVW5Srr+OQyChrQF25ET7i2KiugdgfgXDSFqeUk/D5gaKDGY2XFtU75gmY0DqgoWAYo4eGCv4ZzYpHRW2HtGxAqJ4aAzSLc4kdVd+F2JtKAdRFXL15G558NWidA7eSQmTPu7gmhJVhblVX1brYr8ytYSefQEqjDrh61DAcZlPBwgB0/qDbZXl2MI5LMIljvx54aW2EBtS9ZTefyeH2aHTvYp0lR7JUyGLPZ+0NlXYpxm7xNXTQqx8oOl3OG6hoe03DQniqriFDCwwFVBaZsorWjFRZQW8Nzdkpvhd3dvUzeAAAgAElEQVTaEsD2LdW3wjJUtjq4Gc/g6qK9rbCq6i00TTMWcTu62QC1NTx23GPHUPWdYozaXGUVEUp4OKDqUY39E6262oFRS2dILa2wDJUrPOxmZjtbYX1QU29x7UYCS/EMAnV+7OurvcoKqLtBy+QKGJ8qXutiw5yluobHzs4/UaGEhwOqlojN3ZJdOgv9o4o7y1Gbux28ESv7JlpVdSns2GF/fwjB+jpbfqaqG7TXZ6LI5AroaG7AjhoMBxlmnGr+UcIRSWRxZX4FgLot6QAlPFxQsUS8ks7h0hwzHLS5wqNOmAzstmf3RKxqvKrEiqodgDySQ1XfLX5VVsUCBWCs6IG1Y2sztihoOMighIcDKpY+xyf0Vtj+9iZ0h2pvhQWsd9OoEycAmIkkMRNJ6a2wg7UZDjJMgbdascoXNNsFy4C6egse9xyp6vFkdVi2A7/Cq6WdWieRUfj/QvdQccdkd9swoGacAHMXvrsnVLPhIENVI73L8zHEM3m0BOpwd7d9rbAq7sYTmRwuzOpVVlurYYo6no/Y7Bqs4jvFMOZ3m/SZokIJDwd8Cu4ueQjaVNWlGLslGxclZTUpRXNGu1thVdSljE9GkC9o6Ak1ojfcZPvPVylW81bDQZsvwVStyaJgMRy0y+pAVCjh4YBqGh7dcJAqPJuFy7GDoh4gPGIFqOnxxEPrBKg5DtkG7e7uNrTWaDjIULXC88bCCmKpHJoDdditqOEggxIeDqg2MN66kcDNRBaBej/29dmjSQHUSwwB3XDw7LR99xwxVL1olZf3h1XDo8r7ZedVJVZUHIejNl0YakXZ7ttirA4MhFFfp3ZKoPZ/nUuo5mtht+EgQ7U4AcDr03or7JaWgC2tsAwVNTzLiQzeWNANB+1uhbVe76LCOq5pGsYmeCWH6iXTozYLlgE1E0PATKRVbkdnUMLDAdU0PMwYzu7zXRV1FsZtw4PttrTCMlSMFTOyvL2zxfZWWGvoVYjZxFISiysZNNT5bK2yAurpw7L5Asan7DVJBdS9S4vN76p3aAGU8HBBtSsTjFK6zQp+NXeWvHfh6gRr1OY7oaxYk00VIsaOHfb1hdHYYI/hIEO1Suv5mShS2QLCTQ24fWuLbT9XRbuRaCqLy4bhoNqCZYASHi6otDjF0zlcmNU1KfZrB9SJE8PO2+StKJkcss4QDq2wfsUqPDzvOTKOapRIDUtdzv02dv6pOAbHri9D04DtW5rR2Rp0+3G4QwkPB1TaMZ2Z1A0He8ON6AnbYzjIUK1EPGdphT1g9/GfYrEqlBgO8ljE1dLw8LznSLUuLV6dfypqeHhonUSGEh4OsKlWhR0Tr4oFoJ52gO3C7WyFZagWq8vzK4il9VbYXTYaDjJU0vAkM3mcnylWWTlUw1Tzw+Jhkgqot+kArFYH6ut3AEp4uKDSjolvKV2tErGxC+dyRKOWfmCUcyusShWe8cll5AoautqC6LO5ygpYKtIKTFgLsTQmlpLw+YBh26usao3BQsH0V/OCYBmghIcLqpQ+NU0zu444ltJVqIQBfO+j8dvY8SUCvI4dGCpVeJjW6V6bLsFcjUr+MkaVtasNbY0Ntv5s1TZoVxfjiKZyaGzwY5fihoMMSng4wIRysluQX19KYCmeQaDOj/39Idt/vkreMplcAeNTEQB8zsNVO9LimUgDpQmi7O8Xj6tKrKg0Ds0qKz9xtzpjkFVZ29GguOEgwxv/lS4he+XCaIXtDyFYb28rLABD7KTCBHJ+RjccbG9uwO2d9rXCMsxjB9t/tONEkllc4dwKa62DyFxp5V1lBSyaQ4njxDD0O4MctE5Qq8LjteMsgBIeLqii4TEcODlMHoBaJWJzorXXcJCh0u6SXVR421Z+rbCqVHgmbyaxuJJGvd+He/rtNRxkqKIPy+ULGJ+kCs9msbbvewVKeDigysAwHDi5ldKtwlK5Y8WzbRhQS2cxwtFwkGHNOWV+t0zDwZDthoMMVe5puzAbQypbQKixHjs7W23/+SpdSBtLZXFxLgaAKjxEjahQuUhkcjg/w3dAlJrDcfkVjsHjskIrqgjhAYsIl2MrrM/nU6KNeJTzcRagjoaHJYfD2ztsNRxkqLKRBYAzExFoGjC4pQnb2tQ3HGRQwsMBFXwtxicjyBc09IQa0dfexOV3+GA9dpA3VvOxFCZv6q2wQ4N8jh2Y0kL2RcnJVlgVtCk8bSEYqgjieZvoqVIJAyzvFSe5gqhQwsMBFZyWnTjf9VnePpknW6Z12tVtfyssQ5Xd5RsLK4gVW2F3c26FlV1Ll8rmcW6az7UuVlS54oW31YEqYxCwxso7+h2AEh4uqDAweE8egDrmcMZt8hwnDxWOSQEzkT4w0M7FcNCK7D5Pr01FkCto2NYWxEAHnyoroEblYnEljWs3EgCAIU7aMBXiBOiJrRPHyiJCCQ8HZF+cNM1y7MBJsAyUanhkjRUAjF5zQGdRHKm0C988smt4Rjl3/jFU0PCwe9nu6mpFuImqrOtxdTGO5UQWwXo/dvfY768mMpTwcMDQ8Ei6s9RbYTNoqPNhXx8vTYoaGp5svoDxKb4dWoAZK5kXJcDZUrqR8EgaNHZUynsXroKGx4lEWpUKj1llDSNQ760UwFv/tQ4hu0kcmzz29oW5tcICatj/X5ixtsLabzjIUGFRiqayuGwYDvKv8MhcadUNB52phqmg4eF1YagVVSo8TlZZRYMSHg7IPjDMO6H47sJLNDxcfxM/Rizt6DxaYRkyL96MMxPLjrbCyqzhmY6kMB/jazjIkN3jSTccLF7rwrEapsIYBCw+WB4TLAOU8HBB9u4QJmjjvQsv0fBIXg3jvwvXP8qaRAOWIxqHdpYya3jYorSnN4SmAL8qKyB/9fDiXAyJTB5twXrcuc1+w0GG7HECgJV0Dpc8aDjIoISHAzLXLVLZPF43WmH57gCsQkxZJxGn7NlV0A9Yr99wAvZ2yfhuOap1klwfxlzOh7e3c62yyh4nABifWEZBA/rbm9AVanT7cRyHEh4OyFzhGZ/UW2G72oLo52Q4yPBLruFZXEnj+lICPp8+2fJE9t1loaAZd2g51QrLFj8ZtSlOOCwzZK8e8nY5Z6jQKemE1klkKOHhgMyLk3VA8GyFBUorPPJFyjx2uKurFSFOhoMM2XUWVxfjiCT1Vtg9vc60wsqqudANB4uaFCcSHuPeEu6/igu8HZYZso9BwBor7x1nAZTwcEHmCo9Ttv8MmXeXhtbJAXt2meMEmIn0gYEwGjgbDjJk9Zc5Nx1BNq+hszWAwS18q6yA3Bu0pXgGby7GAfAfh7KPQS8bDjIo4eGArHdp6a2wzg4ImbUpRjcbR3NGBtMPyBgnwK2dJdt4yBU0Ju4eHuzgXmUF5NamsA3aHdtaEG7mW2U13icZAwXgrRsJLMUzCNT7sdehKqtoUMLDAVlL6ZM3k1hwqBWWIeuuqaQV1pFjB/2jrJOtUzoLK7K+W+yqEicSaUDeOAHOJtLGyZ98YQJgjsF7+r1nOMjw5n81Z2QtEbNy596+EFfDQSuyVnguzMaQzObR1liPOzi2wjJ8EusHYqksLhqtsM6JJWXdeDjdvi9rnABLN5sDFWnZNTxevTDUCiU8HDA1PHINDdNwkHbhG8F2S8ODfFthGbLGCdA7/zQXWmFl3JFPLycxG02hzu/DgQFnq6yyHcHnCxrOTDhjCwHIO68znE6kRYQSHg74JJxoAeuxg3M7AFm1KSMOa1Jk1lmYWidnJ1qfhAsUO6LZ3dOG5kC9M79U0iaLS3MxxDN5tAbrcVdXG/ffJ2vlHgASmRwuzOr+ak4eK4sGJTwckLFErLfCMsNBqvBshJOldEDeXTjgXildxgXKjXuOZB+Dw4PtqHOgyiqzc/eZiQgKGtAXbkRP2HuGgwxKeDgg4wRydko3HOxsDWKgg38rLEPG5PDGShrXbiQAAMMDDrkGSxgnoLQV1umdpYyaCzORdkHr5NhvtAd2RONURdq4+0+2QMHir+bRdnSGMAnP008/jR07dqCxsRGHDh3CK6+8su7XLy8v45Of/CR6e3sRDAZx991340c/+pFDT7s+MpbSrbtwJ1phGTLuwtmxw51drQ60wurIqh94czGO5YRuOOh0K6xsVbF0Lo9zU8VjBwe8nRiyxYlhdLM5LO6WbQwCFuduh651ERWHDonX54UXXsDx48fxzDPP4NChQ3jqqadw9OhRXLx4EV1dXbd8fSaTwbve9S50dXXh+9//Pvr7+3Ht2jW0t4vxfyZLF2QaFk5a2VvxSagfcOOIRtZyOtM6udEKK9u7dW46iky+gC0tAdy2tdmx32vESaILfJcTGVxd0A0Hh526m03CzRlQrLI6fAQvKkIkPE8++SQeffRRPPLIIwCAZ555Bi+++CKee+45fPazn73l65977jksLS3h5z//ORoa9B32jh07nHzkdZHNaVk3HHRHZyHj7tKN5FDGOAHu3t1jLFCSDESzS5KqrBvBxuDOzhZ0tAQc+Z2ybjquLyVwI55BoM6PfX3eNBxkuH6klclk8Oqrr+LIkSPG5/x+P44cOYLTp0+X/Z5/+7d/w+HDh/HJT34S3d3d2L9/P5544gnk8/k1f086nUY0Gi35wwvZLpmbjqQwF00XW2GdTnjk0g/k8gWcmXS+vVO2ODHcvLtHtpiR1mnzuGNkabn7T5K5HTA3Hfv6QwjWO+OvJiquJzyLi4vI5/Po7u4u+Xx3dzdmZ2fLfs/Vq1fx/e9/H/l8Hj/60Y/whS98AV//+tfxl3/5l2v+npMnTyIcDht/BgcHbf3vsCKbhodNHnt629AUcHZAyBari3MxJDJ5tAXrcVcXf8NBhnnsIEecAGAlncPFYiusG6V02ZoHRq+5Uw2TLU6AxRbCBXE3IFfzgNcvDLXiesJTDYVCAV1dXfjbv/1b3HfffXjooYfwuc99Ds8888ya3/PYY48hEokYfyYmJrg9n6HhkWRQuGlIZR47OP6rq4JNHkMOGQ4yZCynj08sG62w3Q4aDjJk8niajaQwHUnB7wOGHK6yyhQnQDccHHPw4l6GdbTLlRw6b3UgKq5reDo7O1FXV4e5ubmSz8/NzaGnp6fs9/T29qKhoQF1dWY1Ys+ePZidnUUmk0EgcOuZbjAYRDAYtPfh10A2DY+bA0K23aXbWidZ4gRYjmhcEkrKpE1hVdZdPSG0BJ2dlmWKEwBcmV/BSjqHlkAddvXwNxxkWCs8ssztiUwO52f0a13c0NGJhusVnkAggPvuuw+nTp0yPlcoFHDq1CkcPny47Pe8/e1vx5UrV1CwlAUuXbqE3t7essmO05jeMuKPCt1wUL8E040B4XdQnGkHhmDZ4UVctjgB7lxVYkUmjyc37zmSKU6AGashhwwHGT7LaqlJonh6bTKCfEFDT6gRfe3O+auJiusJDwAcP34czz77LP7+7/8e58+fxyc+8QnE43Gja+vhhx/GY489Znz9Jz7xCSwtLeFTn/oULl26hBdffBFPPPEEPvnJT7r1n1CCTLvxc9NRZPMatrYEsH2Lc62wDJm8LW7GM3hzUW+FddrPQjatU6nhoDs7S+OGeQli5vRVJVZkmq8AM5F2Xuskn4bHDa2TyLh+pAUADz30EBYWFvD4449jdnYWw8PDeOmllwwh8/Xr1+H3m7nZ4OAgfvzjH+PTn/40Dhw4gP7+fnzqU5/CZz7zGbf+E0qR6C4ta7eDk62wq5GhRMyMznZua0F7s7OVRNkWpWs3ElhyuRVWFm1KJlfAa1PuVVllc/F26wjeWkySZRwathAOap1ERoiEBwCOHTuGY8eOlf23l19++ZbPHT58GL/4xS84P1V1yFS1cNMnBZBsF+6muFuyy0PZe7XfxVZYWZLE12eiyOQK6GhuwO2dLY7/flniBACRRBZvFA0HHTdJhVwaHt1wkCo8VoQ40lINmc7E3W5ZlCpWE+4lh36JqoaANZF2b2cpS+XCPKJxp8oqS5wAcwzu2NqMLQ4ZDjKs/9fIoM+cvJnE4koaDXU+7OsLu/04QkAJDwdk2THNRJKYiaRQ5/dhaNCdASGLwDtf0DDmps7CL0ecGG4n0oA849A8dnCpyipVRdp9I0tAjgqPYTjYF0Zjg7cNBxmU8HBAlkt12RHN7p42NAfcOd1kU4joE8iluRjimTxag/W4u9u5VliGTK3DiUwOF2b1Vlg3S+my3KVlHju43b7vyq+viFEXb/32S1bhMa/AoeMsBiU8HJClo2bUZf0OIM9CbhoOhh1thWXIpOE5M6G3wvaGG9Ebdq8VVob7x+ajKUwtJ3XDQdcqPPpHkeME6C7jzHDQjfZ9n6QVHjIcNKGEhwNGiVhw92ARBoQsGh63ux1k0vC4qXWyYmhTXH2K9WHv1d3dbWh12HCQIUOcAODKwgpiqRyaA3XY5UKVVaYKTyqbx+vT7l3rIiqU8HBAhh1TOpfH2anigBAi4RE3VoAlOXTpiEaWOAHudrNZkUHDYx47uD8GRY4TYFakDwyEUV/n/NIlU4XntakIcgUNXW1B9IWdv9ZFVCjh4YDh/+Hyc6zH69NRZPIFbGkJ4LatzhsOMmTQDywnMrjKWmFdqvDIcvSnt8K636EFyHEM6KbDMkMWHZ0IibQMm1mg1OXcTX810aCEhwMyLE6s2+HgYLurA0IGvRNzDL69swUdDrfCMmQR4F5fSuBGPFNshXXHcJDBPJ5EXZwyuQLGJ3XDQTePHUSPE0MkqwPRx6HbFWlRoYSHAzJcHmoOCDGOHQQOFUZdsrK3IkOcAPOIRoRWWNH1YRdmo0jnCgg3NeD2rc4bDjJEjxMARJJZXJ5fASDKOBQ3WJqmmRtaEiyXQAkPB2TYMY1ZKjxuIoN+wE3vD4YMcQLEEMIzRK8eWu+E8rvQ+ccQPU4AcKZYZb1tazM6W4OuPYcMFZ6p5SQWYmnU+324p58MB61QwsMB0e/wmROgFZbhE/xMPG9phRViZylonBhuX1ViRXRtigiJNGCNk6CBgvvmjAxDCC/qSwXzvdrXF3K9yioalPBwQPTuELaz3NUTQotLrbAMn+At/FfmV7CSdq8V1kDwOAFAMpPH+RlmOOh+hUf0cShK+74MR/BumzMyRN/MAqVXlRClUMLDAdHLniLtwkXXprBYDQ20u9IKy5BBOzA+uYx8QUN3SIxWWOMqAAFDthBLY2IpCZ8PGBakaiFinAC9mjIqyFGpDONwVICKtKhQwsMB4XeWgpTSAfG1KSK4UQNy7MLNzj8xWmFF1qYYhoNdbWhrbHD1WUSOEwBcXVxBNJVDY4Mfu3pcrLJC/HGoGw4WO/8EmN9FgxIeDvgE3llmcgWMT7EB4f4OwNhcCjrZiqKzkEHDMypYK6zIHk8i3XMkuo0G8985MNCOBherrID4sTo7FUE2r6GzNYiBDveudREVSng4IHKF5/WZKDK5AtqbG3B7p3utsAyRd0yRRBZXBGiFBcQ30bO2wrqdHDJEHocidbOJPAYBU+skQqyMazgEfKcAa/XeXX81UalJsZrNZjE7O4tEIoFt27Zhy5Ytdj2X1Ig8gYxauh1EGBBml5a7z1GOsUmzFXari62wgPg7y8mbSSyu6K2w+wVphTX8ZVx+jtVk8wWMTzIRrvsVHuF1dNfEqYb5Ba4aAuL4q4lKxRWeWCyGb37zm/hf/+t/IRQKYceOHdizZw+2bduG2267DY8++ih++ctf8nhWaRB5cRJvFy6ufsBqz+42opvDsYlWpFZYUe8fuzgbQypbQKixHjs7W91+HGHjBADRVBaX5oudfzQO10WvsoozZ4lIRQnPk08+iR07duDv/u7vcOTIEfzwhz/E2NgYLl26hNOnT+PEiRPI5XJ497vfjfe85z24fPkyr+cWGpFvHzYWcUF2AMykUciER4B7jhgixwkQ4xLMWxDUM4W9V8PbO1w1HGSIvEEbn4hA04DBLU3Y1uZulRUQW+A9E0lhLkqGg+tR0ZHWL3/5S/z0pz/Fvn37yv77Aw88gN/5nd/BN7/5TTz//PP47//+b9x11122PKhMiCownS8aDvp8+o3DIiCqr0WhxHDQ/UVc1DgxRCyli3q0bFYO3U+kAbG9sEzDQTHeK5GTQxarPb0hNAXEqLKKRkUJzz/+4z8a//ttb3sbXnrpJYRCt14Q2NjYiI9//OO1P52kCDvRFnfhu7rdb4VlmA1tYgXrjYUVxFI5NDXUYbfLrbCA2BOt3gobBeC+E64VUbUpot1zJGqcALGqrIB1M+vuc5TDvE1ejFiJSNVdWr/4xS+QSqVu+Xw0GsVnPvOZmh5KdkRdnEYFuG14NX5Bd5dsoj0wEHbVcJAhsnbgtakIcgUN29rEaoUVUZuyuJLG9aUEAPcNBxkixgnQn0cUh2WGyOPQdO4WI1YiUvFM/oEPfABf+cpX4PP5MD8/f8u/x+NxfO1rX7Pl4WRF1OMHkbw/GKK2Dos60YoWJ6D0iEaEzj+GiHdEsffqrq5WhJvEqrKKFCcAuLoYRySZRbDejz29t54kuIGo4zCdy+PclF5lJcHy2lTclr59+3b8+7//OzRNw9DQELZu3YqhoSEMDQ1heHgYFy9eRG9vL49nlQarDlHTNCEWgWy+gPEpsTq0AKuvhcsPsgpRLitkiNy+PyJg5RAQ84oXEbtoRPV4Yon0gYGw64aDqxEt4Tk7FUUmX0BnawCDW8SpsopGxQnPk08+CQAIBAL4n//5H0xPT2N0dBRjY2P4wQ9+gEKhgK9+9au2P6hM+C0JTkED6tzPd3B+JopUtoBwUwN2CmA4yBDxbppoKovLRcNBcSo8+kfRJloRDQcZIuotRLmqxIqoTRbsTiiR3iuzW9Ld51gNe6+GBbnWRVSqNh6Mx+NoaNBLsr/5m79p2wOpQGnCo6EO7r+ArJQ+PNguRCssQ8Rd+Nj1ZWgasH1LMzpdNhxkiGp1MLWcxEJMzFZY0Y4fcvkCzkwUr3URJJEGYMwHgoTJQMRbv825XaxgmUfw4iTSIlJ1nZAlO0QZSo603HsMKyKW0gExKxdiap3EWrwZrLojYiusT7DKxYXZGJLZPNqC9bhzm/uGgwwRx+BKOodLc8xwUMRx6PKDrELU+V00Kkp4rl+/XtEPn5qaqujrVcFaQBFlEhkR7GJHhoj6AREnD1OA6+pj3MKoYG3DVkSrHrIjmuHtYlVZIeAYHJ9YRkED+tub0BVqdPtxDIxxKFCwZiJJzERSqPP7hPFXE5WKEp5f+7Vfw+///u+ve3VEJBLBs88+i/379+Of//mfa35AGbEeaYmQ7yzE0phY0g0HhwQR4TLYmbgQgYI+kY0KmPCwOIlSrWCMCNbNZkU0Dc+ogEc0gJg6OhGNLAFrR5u7z2GFVaR397ShOVDT9ZjKU1F03v/+96O1tRXvete70NjYiPvuuw99fX1obGzEzZs38frrr+PcuXO499578dWvfhXve9/7eD230KzW8LgNW8Dv6mpFSBDDQYZou/Cri3FEUzk0Nvixu9d9w0GGiN1suuGgrkkRxQnXimjHgKKZ6DFE9MIyzBlF26AZWjox3ilArDv/RKeihOcf/uEfMDExgS996UvYtm0bent7sbi4iGQyic7OTnz4wx/G0aNHsX//fl7PKwVWkbwIw0LULhpA3EXpQH+7UK2wosUJAM5NR5DNa8K2worUfbQUz+CtG7rhoGjJoWjGg7rhoJgVHhGNB0cE7PwTlYoSnr6+PoyNjeHo0aNIJpN44okn0NXVxevZpMUnmIZH5AEhmjbFaBsWTuukI0qcANPK/uB2MVthRaoesvfqjm0tCDeLVmXVP4oQJwB460YCNxNZBOr92CuI4SBDNJPGdC6Ps9NkOLhZKtrC/vEf/zEefPBBvOMd74DP58N3vvMd/PKXv0QymeT1fFJSouFxuUycyxcwPilyhUf/KMru0ujQol34hoicSANi3dMmohCeIdoizo5o7ukPI1AvTpUVECuJBoDXp6PI5ArY0hLAbVub3X4c4anobfrDP/xD/OpXv8J73vMeaJqGp59+GocPH0YoFMKePXvw27/92/jKV76C//iP/+D1vFIgkobnwmwMqWwBocZ63CFQKyxDpBJxLJXFRdYKK1iFRzQBLmDx/hBwEQfEaiEeFezCUCumLkUMRNU6AeJt0KxaJxGrrKJRsaT7wIEDOHDgAJ5//nmcPn0aLS0tGB8fx9jYGMbGxvCv//qv+PKXv4xYLMbjeaXAL5CGh00ew9s7BGuF1fEJpE05MxGBpgEDHU3oahOnFRYQK04AML2cxGxU7FZYUfxl8gUNZybENYYTrXoociIt0gYNgLBaJ1Gpuoft8uXLxv8+dOgQDh06ZPxdlIHjFj6BKjyjgnY7METSD4h4mzxDuGOHYqz29IrbCitKZ9vF2RjimTxag/W4q0uczj+GSGMwns7hwqyuSaFxuDEimqSKDJcDUiqtiTMwRPWzYIiyCwdEL6WLsXgzRNU6WRHFaXl0Qn+vhgbDqBOwyirSGDwzqRsO9oUb0RMWq8oKiKXhmYumMLWchN8HDA2IN2eJiFiKMIUQYYFaXEnjWrEVdljQCo9fkORY0zQhLytkmOZwYiCqc7cVUTQ8rJtNxPcKEKcSBlgSacE3aG4n0YBZkd7VE0JLUMwqq2hQwsMJEUSmY8XJ486uVoSbxGqFZRg7JpdXpTcX41hOZBGs92OPYK2wgFgannQuj3NT4rfCilK5ENG524pIHk/GsbLgGzS3k2jA6q8mZqxEhBIeTph3RLk3MkQ+omGIoh9gk4eIrbCAOEekAHB2KopMXm+F3b5F3FZYNgbdDNnNeAZXF+MAxK2yGvd/u/xqaZom9FUlgDVW7o9DcliuHPFmdkUQYYES2fuDIcouXHytkzj2/9YLQ0XW64lw/DBWPCbd2dmCjpaAa8+xHqJUeK4vJbAUzyBQ58e+PvGqrIA4FZ5MroDXporXugi8oRUNSng44baGRzccZANCzGs8TmUAACAASURBVEUcEMcDZFTw8rBIWleRPWWsiCAwHRG484/hE+D4HTBjtb8/hGB9nbsPswaimFmen4kinSugo7kBt3e2uPosMkEJDyfc1vBcnIshkcmjLViPu7rEMxxkiOABspLO4aLArbCAOLtwQHyHZYYIMZNC3O13P05A6VUloiJKhceaSItcZRUNSng44bbIlJ2FD29vF9JwcDVuTrbjE3orbH97E7pD4rXCWnF7UZqJJDETSUnRCuu2Pkw3HBT3NnmGKBoe1r4v8hG8MFYHgvuriQolPJxwW8Mzek3sbgeGCDsmGSoWIsQJMCfa3RK0wrqt4bk8H8NKOoeWQB129YhnOMgQoRKWyORwfkbMa12siBArQHzNoahQwsMJtxco5ikjqp8Fw+2jP0AOTYq/OFLd3oWzzhCRk0OG2/4y7IhmaLBdSMNBhggeT+OTEeQLGnpCjegNN7n4JOsjgt5pPpbC5M0kfD793SI2DyU8nDDnN+dHxlI8gzeLrbD3ClxKB0z9gFu78FLDQXEnDxG0TgCENmdcjdu78VEJKoeA+8fvgKVpQODqDuD+RhYwE+ld3W1oFbzKKhqU8HDCzQ4RNtHesa0F4WYxDQcZLC90a7J960axFbbej319Yl6CCbgfJ6C0FVaGUrrbGh4ZbCGA0qqFWwm1bLFyNTmcEL/zT1Qo4eGEm/4yMhzRMNxuHWbJ4f6+kJCGgwy34wQA56YjyBRbYXdsFddwkOGmhmc5kcEbC3qVVfRxaL3exY11XNM0aaphIlRaR6+JX5EWFXFneMkxr0xw/nfLslsC3NfwyBIrtwW4QGkiLUMrrJseT8xwcMfWZmwR1HCQYZUXuRGriaUkFlcyaKjzCV1lBdyfr7L5Asan5NnQigYlPJwwhYDOjgy9FVaO83DAfZ2FcbGj4Ec0bhtZAnJcVWLFTW2Kec+R2O8VgJLk1ZWKdPGIZl9fGI0NYhoOMtyutF6YiSGVLSDc1ICdZDhYMZTwcMKte3wuzcUQz+TRGqzHXV3itsIy3PS1SGRyuGAYDoq9iAuhHZBoEQesuifnf7dxRCN4Ig2Y7xbgUnIoU+df8aN7/mpmrGTwVxMNSng44ZaGhw2IocGw0K2wDDd3TGcmIihoQG9Y7FZYwP3ukLloClPLSfh9wAFJWmHdGoOFgoYxiYzh3NbwyFQNc1vDYyQ8gnffigolPJxwayE3jmgkmDwA947+AHn0O4D7d/iwisXdErXC+l0ymLmysIJYOofmQB12C2w4yCjR8Dgcq2Qmj/MzepVV9GNlwOKH5dLvl6V9X1Qo4eGEaRTn7NAwWxblGBBuVi5MEa74sXK7wmPswiVYlBhuaXjYEc2BgTDq68SfYv0uanhem4ogV9DQ1RZEX1jsa10AazOK8wNxcSWN60sJ+HzAsASVQxERfzRKCtPwODkulhMZXGWtsJKUPM17fJydQEpbYcWPldt3+BixkmiidUtvIZMtxGrcOoK/V5LOPzd1YSyRvrurDW2NYvuriYpQCc/TTz+NHTt2oLGxEYcOHcIrr7yyqe/73ve+B5/Ph9/6rd/i/ISbx402YjbR7uxsQYfgrbAM46Zmh9v3ry8lcCOeQaDOj/39IWd/eRW4WeHJ5AoYn5THcJDhVsxkOioFVld4nP3doxLcJm/Fza5S47ogCSrSoiJMwvPCCy/g+PHjOHHiBEZGRjA0NISjR49ifn5+3e9766238Cd/8id4xzve4dCTbg43Jls2eQxLNCDc0qaw5HBvXwjBerFbYQF3KzznZ6JI5+RrhXXDMyWSzOLy/AoAeRamkt4GB2OlaZpxVCpLNczNPhBW4ZElkRYRYRKeJ598Eo8++igeeeQR7N27F8888wyam5vx3HPPrfk9+XweH/7wh/HFL34RO3fudPBpN8aNBUqmbgcG7cI3h5sVHmsrrAzHDgw3OmqY4eD2Lc3obA069ntrwS0Nz+TNJBZiadT7fbinX2zDQYZbFZ5c3lpllSORFhEhEp5MJoNXX30VR44cMT7n9/tx5MgRnD59es3v+4u/+At0dXXhd3/3d514zIpwuksrX9CMyVaWRRxwT2cxIlkpvWQT7pImRab3CoARNCffrVHJzBkB93x42BHNvr6Q8IaDBi7dz3ZhNoZkNo9QYz12drY6+8sVQoj+0sXFReTzeXR3d5d8vru7GxcuXCj7PT/72c/w7W9/G2NjY5v6Hel0Gul02vh7NBqt/oE3gdMansvzMawUW2Hv7pZnQLjhIKy3wsYAyLOIr9ZZ1DlYaJGtGsZwoyomczcb4HCsrsnTNMBwq8JjyhU6yHCwBoSo8FRKLBbDRz7yETz77LPo7Ozc1PecPHkS4XDY+DM4OMj1GZ2ebNkufGigXYpWWIYbR3/jk8vIFzR0h4LolaAVFlhtDudcrOZjKUzeTMLn080sZcJpG55CQbNUeORZxAF3/LBkuTDUilt3aZlyBXliJSJCVHg6OztRV1eHubm5ks/Pzc2hp6fnlq9/44038NZbb+HBBx80PlcotvnU19fj4sWLuOOOO0q+57HHHsPx48eNv0ejUa5JD9s1OTWBGII2SY5oGK7uwiVphQUAnyWHdVYIr8dKxlZYpzU8VxdXEEvl0Njgxy4JDAet+H0+FDTNsYU8lc3j3HTRcFCi5NAtp+URiSw0REaIhCcQCOC+++7DqVOnjNbyQqGAU6dO4dixY7d8/e7du/Haa6+VfO7zn/88YrEY/uZv/qZsIhMMBhEMOicidNqvQdZjBzfuiJIxVta0zJVYSZZIA86/W8zl/MBAOxokqrICzsfqbNFwcFtbEAMdYl/rYsXngobnxkoa124kAJDhYK0IkfAAwPHjx/HRj34U999/Px544AE89dRTiMfjeOSRRwAADz/8MPr7+3Hy5Ek0NjZi//79Jd/f3q6/CKs/7xbMadmJCSSSyOKNouGgbAPCaQ2Pbjgon5+FW/cdjV5jd0LJkxwyTFdcZ36fjIk0Q4+V5vgG7eCgXJ1/brh3s/nqrq5WhJvkqrKKhjAJz0MPPYSFhQU8/vjjmJ2dxfDwMF566SVDyHz9+nX4/fLsmpwsfbLrJHZsbcZWSVphGU5rByZvJrG4kkZDnQ/7JWmFBVYlPA7FKpsvYHxK3rt7nH63ZEykGU43WRh3/kkk7gbc0fCMSKh1EhVhEh4AOHbsWNkjLAB4+eWX1/3e559/3v4HqgGfg5ULGf13GG7twvf2heVphcXq1mFnfueFmRhS2YK0rbBO6sOiqSwuzcvV+WfFyUqrbjgoZzXMDQ2PtLYQAiJPyUQynNTwyNjtwHBaO2DswiU7+nPDK8UqlJSxFdbJe9rOTCxD04DBLU3Y1iZXlRVw1g9rOpLCvGSGgwyntZm5fAFnJuWshokIJTyc8Du0kBcshoMyKvid7tIyRbhyxarkSMuhapjMiTTgrPmnmUjL9V4xnByHrKN0T28ITQF5qqyA8xqei3MxJDJ5tAXrcec2+aqsokEJDyecKn1eWdBbYZsa6rBbslZYwHo3Df8JJJXN43WjFVauRdwNDY/MR6WAs7qUEQkdlq046Yc1KrGnjDGvO/T7WKyGt7dLWWUVDUp4OOHUmTjbhR8YCEtlOMhwchf+mqUVtr9dnlZYoPTSQiditbiSxvWlBHw+uS6jteJU1UI3HJT72IEtpo5UeCT2lHGqcs+QOVYiIt8KKQsO+TXI2u3AcFI7YN42LFcrLLDa/t+5Xfid21oRksxwkOFU1eLNG3FEklkE6/3Y3RPi+rt44ZTeSTccLF6CKeEibr5Tzvw+mTv/RIQSHk44tROQtduB4ah2QPJYOSnwlj1WgHPvFkukDwyEEaiXc0p1KlbnpqPI5jV0tgYwuEWuKitgiZMDE9bNeAZvLur+avdKqg0TDTlHpwQ4cclcJJnF5fkVAPLuAJi1Eu+dpd4KK6+4G7DoeBwUlsrov8Mww8V70yG31glw7ioc4xLMQXmudbHic1DDw/zV7tjWgnCznFVW0aCEhxN+BwbzmWJ31vYtzeiUzHCQ4ZTWaWo5iYViK+yBAblaYRl+h45Jc/kCxif1YwdZk0PAuhvn+3tk72YDLO8W51jJfFUJ4KyGx5ArSDwGRYMSHk44cfwge2eIFf5Hf/rksbcvJJXhoBUfnGmJvTAbQzKbR1uj3K2wTozBlXQOl+bkNRxkOHVcKruJnpMaHlbhkXnTIRqU8HDCCQdh2Y9oAGeO/gDLLlwyw0Erzi1K7NhB7lZYJ6qHZyaWUdCA/vYmdIUa+f0izjgRq5lIEjORFOqkrrI6M1/lCxrGjM4/eecs0aCEhxO8S5+FgoYxhYSlvHdMI5K3DQPOx0rmRBpwRsPDtE4yH2cBVn8ZnrHS36vdPW1oDgh1q9GmcerKoEtzMcQzebQG63FXl3z+aqJCCQ8neBtUXV1cQTSVQ2ODH7t75R0QTlzGpxsOytsKy3Dq4sJRRY5Kneg8Gp2Q+4iG4XNAHzaqxAZN/+jU0d/QYBh1EldZRYMSHk7w9rVgu6UDA+1okNBwkOHEMc256UixFTaIgQ75WmEZTpTTb6yk8daNBAB5r0lg8PZ40jTNXMQlrhwCDmsOJT6icczqQIHkUETkXSkFh7eDsClok3fyAJy5m4YlhwclNBwswYFFie0sVWiFZfojXuF6czGOm4ksAvV+7O2V03CQwfsqnHQuj7NT+rUuMifSThk0UsLDB0p4OMG79KlKy6ITd9OoMnk4ESuWSMseK4D/XVosObynX17DQQZvfdi56Sgy+QK2tARw29ZmPr/EAXgn0QCwnMjg6oJuODgscZOFiMg9SgWG5wQSTWVxaV5vhZW9wsPbW0Y3HFRFk6J/5Lm7lP2qEiu8q6yqvFcAfw2P9cJQmausThz9MV3Yzs4WdLQEuP0eL0IJDyd43uMzPhGBpgEDHU3oapO3FRbgf9/RTCSFuWgadX4f7pG0FZbBWz+QyxdwZlKNyiHAX8OjgsMyg3+s1PCUcUQIf02NWIkIJTyc4DkwVDmiAfhreFis9vTK2wrL4L27vDS3gkSxFfbOLnkNBxk8q6zxdA4XZ4uaFAXGIW9B/Kgi7ftOaHhG6MJQblDCwwmei5NKpXTe2gFVtE4Afw+QEYvhoAqtsDyFuGcmdcPBvnAjesJyV1kBvve0zUZSmI6k4PcBQwNyz1m8dXT5goYxRawORIQSHk7wWsj1Vlg1jOEA/hoeVbrZAAeE8Aol0gBfXYoxBhXQOgG8Y6W/V7t6QmgJUpV1Pa7Mr2AlnUNzoA67euT1VxMVSng4wWtgXF2MI5LMIljvxx7JW2EB834oHrvwdC6Pc8VWWBV2S7yrYWMKJdIA38VJhatKrPA8WlYpkeat4WGxGhpQo8oqGpTwcILX4sSs7A8MyN8KC/CtWpyd0ltht7YEsH2LvK2wDJ7C0pvxDK4u6q2wKlTDAH6Lk975p043G8B3HMp+YagV3hWeUQXMGUVG/hVTUHgNDNayqM4unF/VwtiFS94Ky+Aaq+LR385tLWhvVqMV1tCl2Ky4uHYjgaV4BoE6P/b1yV9lBfhpUzK5Asan9GtdVEikeWqdALU6/0SEEh5O8JpAWIVHhfIw4MzOUpXk0F8crVyOHZgbtcQuuKvhpQ9jyeG+/hCC9XX2/nCX4OXx9PpMFJlcAR3NDbi9s8XWn+0GPOerSCKLK/MrANSZs0SDEh5O8Dh+WEnncGmOGQ6qMSB4Vi1Uat8H+OoHDIdlhUrpvKqsKnX+GbB3q2Dvjx21+O+oVGXlkfCMFT2wdmxtxhYyHOQCJTyc4KHhOTOht8L2tzehOyR/KyzAb8c0E0liptgKe0Byw0EGLw+QfEEzBMsqLeLG4mRzhqhaIg3wG4eGp4wy4m79Ixd/tWvqvVeiQQkPJ4zjBxtHhlWTogo+Tkd/7DhrtwKtsAxeFZ5LczHEM3m0BOpwd7c6rbA8jpUTmRwuzKpxrYsV7kfwyoi7+XWVGm7UisRKRCjh4QSPhVxFQRu3neU1dY9o7J5sWXI4pIjhIMPUpdj3M8cnI8gXNPSEGtHX3mTfD3YZHhqe+WgKU8tJ+Hz6u6UCPN4pQN8YM8NBVaphIkIJDyfs1vDohoNq7ZYAS9XCbu3AhIoiXD4VHhWPaADT48nOZHpE0bZhHhetsg3aru42tCpSZeWl4XljYQWxlG44uJsMB7lBCQ8n7F6c3rqRwM1EFoF6P/YqYDjI4FG1yOQKeK3YCqtScsirwjOi4FEpwEe0rGI3G8CnyWJUkQtDrZhxsvfnsjF4YCCM+jpalnlBkeWE3SVidkRzT78ahoMMHtqBc9MRZHIFbGkJYMdW+Q0HGTwqPMuJDK4uMMNBdRYmAPD77W0c0DQNYwp2swF8mixUclhm8LpkVTULDVFRZ+UUDLvbrVWcPAA+u/BRS2eICq2wDFMXZmOsikd/t3e2KNcKa7feYmIpicWVDBrqfNjXp0bnH8Nuj6dsvoDxSfWqrH5OK6aqx8qiQQkPJ+xeyFXdAfCoWqh6RMPDSG/0mpqxAuzX8LD3al9fGI0NahgOMuyu8JyfiSKdKyDc1IDbt8pvOMjgUeGJprK4bBgOqjcORYISHk7YuZDH0zlcmFXnEkwrPHQpKt3dY4VLNUyxq0qs2N0BqKItxGpsSw4tibRfoc4/hp1NFmPXl6FpwPYtzehsDdr3g4lboISHE3ZqeM5M6oaDfeFG9ITVMBxk2L2znCu2wvp9wAHF2jvt9gAplBgOqhUrwH5rCBVtIRh2j0OWSKsWKx4VnlGFx6BoUMLDCTvFuKoeZwH8duF3K9QKy7BbF3Z5fgWxtN4Ku0shw0GGVcNTa5KYzORxfqZYZVVIk8Kwexyqe6zMw19NPbsRUaGEhxdsArHhTEvlUrrd/h/GLlzBycNuDc+o4q2wVsF6rev4a1MR5AoautqC6FOsygrYW+FZiKUxsaQbDg4rV2XVP9pZZTXmd8WsDkREvVlOEOzS8GiapvQibrf/h8r30dgeK8U7Q6zSkVpjZo2VSp1/DDv1YUaVtasNbY0NNf88kbD7Lq2ri3FEUzk0Nvixu1e9KqtoUMLDCbtKxNeXEliKZxCo82NfnzqGgwy/oVqu/WdZDQdVrIbZreFRWZMCrKrw1PizVLyqxIqdeifjwlAFx6DdTsum4WA7GhSssooGRZgTfpt2gUYrbH8IwXq1WmEBe0WArBW2vbkBOzvVaYVl2HnsEElkcaXYCjus4MIE2Ffh0TRN6W42wF4Nj8qVQ9vF3QrLFUSEEh5O2LUTYFb2Kk4egL0lYkMoqZjhIMPOWI1N6u/VbVvVbYX126ThmbyZxEIsjXq/D/f0q2U4yLDrCD6XL2B8kh3Bq7eI263hUdVCQ1Qo4eGEXXoLVbsdGPZqB9TehdsZK5W1TgyfTRUe03AwpJzhIMMuP6wLszGksgWEGuuxs7PVhicTCzs3HbFUFhfnYgDUHociQQkPJ+zYMSUyOVyYVXtA2FkiVrmUDth7/Kd6Ig3YV+FRPZEGLBXpGldy9l4Nb+9Q0nDQTg3PmYkINA0Y3NKEbW1qVllFgxIeTthxj8/4ZAT5goaeUCP62pvseTDBMH0taptA5mMpTN7UW2GHBtU+dqiVQkHDmKLGcFbsqvB4QWdhl7+M6iZ6dm7QqB3deSjh4YR5U3PtpXQVz8IZdnnLsIlWxVZYhl1HWm8srCDGWmF71G2FtSaI1b5fqWwe56bVvNbFil3j0KwcqhkrPuJuded30aCEhzO17SzZrd9qTh4ATINGu3xSFE4OzWOH2n6OtRVWRcNBhrUeVu3G42zRcLCzNYiBDjWrrIAZq1o2aIsraVy7kQCgnuEgg11IW2u+Y+38U9FfTVTUne1cplYNj6aZDpwqL+J2lYhHr6mvs7Brd+mVzhA7NDzWXbiKnX8MO/Rh7L26q6sV4SY1q6x2jcE3F+NYTmQRrPdjd496/mqiQgkPJ2rV8EwsJbG4kkFDnQ/7+tTUpACrF6XqgpXNFzA+pbZ2ALBPZ+GVUrodGh7DFkLxXbgd97R5Qetkl0EjM2c8MBBGoJ6WYaegSHOiVlfc0Ql98tjbF1a2FRZYbQ5X3c+4MKN2KyzDDg+QaCqLy0XDQZWrYYC+ONXSRqxf66J25x/DDg2PF2JlV4XHC7ESEUp4OFGrwNT0SVF3twSYZ+JADbGyCCVVbIU1qd3q4MzEsqdaYWvRpkxHUphX3HCQUet8lcsXcGZCv9ZF5WqYHZUwwJzfVa6GiQglPJyo9RZw1e85Yvgsb2C1k+2oR3ZLduwuVXfuXk0tx4BsUdrTG0JTQN0qK1B7RfriXAzJbB5twXrcuU39KmstY3AlncMlMhx0BUp4OGEcP1TxvclMHudn9FZY1XcA9ghL1b2s0IodAm+vldJrEeN6ResE1F65YGNweHu70lVWOyo84xPLKGhAf3sTukKNNj0ZsRko4eFELRPta8VW2K62IPoVNRxkWOfGaiaRxZU0ri8l4POpewkmw18crdXuwq2Gg6onh4xaNDxecFhm1KrhGb2mtv8Ow5YqqwfE3aJCCQ8narmbxroLV7kVFqhdw8OOHe7qakVIUcNBhq9GDc/VxTgiySwaG/zY0+uNVlgj4akwaLrhYFGTovgiDtSu4TE8ZRRfxO3pZvPWsbJIUMLDiVpM4rzQ3smotXWYTbRKmzMWqVkIzwwH+9vRoLDhoJVqr+M4Nx1BNq+hszWAwS1qV1mB2jQ8S/EM3lyMA1B/HNZa4SHDQXfxxqznAqaGp7KBobfCemdAlGh4qvh+o5tNYXNGRq0aHi8l0oxqj5bZLnx4UP0qK1CbuJu9V3dsa0G4We0qa61j8K0bCSzFMwjU+7HXI1VWkaCEhxPVOi1P3kxiwSOtsMAqDU+F1bBcvoDxSe8cO9S6u/SSJoVRrYbHC1eVWKmleuilI5qaj/6K79U9/WQ46AZCRfzpp5/Gjh070NjYiEOHDuGVV15Z82ufffZZvOMd70BHRwc6Ojpw5MiRdb/eaar1/2AT7d6+kNKGgwzr7rnSSeTCbLEVtrEedyjcCsuoRT8QS2Vx0WiF9cYiDpjjsNJ3y2vt+7Xow1S/MNRKrTo6L3X+iYgwCc8LL7yA48eP48SJExgZGcHQ0BCOHj2K+fn5sl//8ssv44Mf/CD+67/+C6dPn8bg4CDe/e53Y2pqyuEnL0+1FR4v7ZaA1U7LlR476JPH8KDarbCMWnaXZyYi0DzYCsvei0o2HjORJGajKdT5fTgwoH6VFai+epgvaDhjaFLUX8Rr7ZQc8cCdfyIjTMLz5JNP4tFHH8UjjzyCvXv34plnnkFzczOee+65sl//ne98B3/wB3+A4eFh7N69G9/61rdQKBRw6tQph5+8PNV2aXlNZ+GrQcPjFXNGhh06Cy/owqxUo7lgi9LunjY0B+p5PJZw+Ks0Drs0F0M8k0drsB53dbXZ/2CCUcsYTGRyuDCr+6t5Zc4SDSESnkwmg1dffRVHjhwxPuf3+3HkyBGcPn16Uz8jkUggm81iy5YtZf89nU4jGo2W/OFJNRUevRXWewOi2t2l15LDWjQ8Xi2lV+Mv4zVzRqD66iGL1dBgGHUeqLLWMgbPTERQ0IC+cCN6wt6psoqEEAnP4uIi8vk8uru7Sz7f3d2N2dnZTf2Mz3zmM+jr6ytJmqycPHkS4XDY+DM4OFjzc69HNRPI2aLhYGdrEAMd6rfCMqrRptxYSeOtGwkA6rfCMph+oNK51toK671SeuVdWl5LpIHqtSle0zoZ71MVIh5D6+SxKqtICJHw1MpXvvIVfO9738MPfvADNDaWz5wfe+wxRCIR48/ExATXZ6qqlG7ZhXuhFZZRza6JaZ3u7GpVvhWWwfQDlU62by7GsZzIIujBVthK3610Lo+zU1Rl3SxeuceOYZz81eLcPeidRFo0hDig7uzsRF1dHebm5ko+Pzc3h56ennW/92tf+xq+8pWv4D//8z9x4MCBNb8uGAwiGHTudmhDw1PBaa+xW/LYDkBP7rSKJpHRieJuyUOTh69K/QDTOnmxFbbSjce56Sgy+QK2tARw29Zmjk8mFtVs0G7GM7haNBwc9sg4rFbDo2maZ3V0IiHE7BcIBHDfffeVCI6ZAPnw4cNrft9Xv/pVfOlLX8JLL72E+++/34lH3TSGhmeT3jK64aD3FnGgut2lF5PDanfhXr67p9IduXUX7sUqayVNFuxetp2dLehoCfB4LOGo1sjy+lICN+IZBOr82NfnrSqrSAhR4QGA48eP46Mf/Sjuv/9+PPDAA3jqqacQj8fxyCOPAAAefvhh9Pf34+TJkwCAv/qrv8Ljjz+O7373u9ixY4eh9WltbUVrq/ueLJVqeKYjKcwXDQcPDHhrYapUm5LLF3Bm0mvagep1Fl6zOrDiq3CBGvHqLryKJgsv+e8wqhV3szG4rz+EYL36/mqiIkzC89BDD2FhYQGPP/44ZmdnMTw8jJdeeskQMl+/fh1+v1mQ+uY3v4lMJoMPfOADJT/nxIkT+PM//3MnH70slZaI2RUJe3pDaAp4a0BUWrm4NLeCRLEV9s4u95Nbp6hmF76SzuEia4X12iKOyhco89Zvb206atHReSlWNTt3eyg5FBFhEh4AOHbsGI4dO1b2315++eWSv7/11lv8H6gGKr1Ly4uTB6Pi5NBiOOiFVlhGNd1s4xPLKBQNB7s9ZDjIqERzMRtJYTqSgt8HDHmsylqpNiVf0IwjLS8t4n5TnFkR/3979xobxXX2Afy/a3u9vq6v2BjbmFuhQLD7cnEglQqtFVpVaVDforQfGiuNqhKRqClS1SRqSz6koqoaNVKKmlSV4EsRaVMRpKqXRCgXqYFQwOYaaKBJfcPYhnrXXrDXuzPvh+Xszu7O7s7yZjxn9/x/EkIxS3NyOrPznDPP8xwGPHKQIoenELly3CJW+YbIdRWubk+ZeyixvvtQ6lJsroRcdsVEUunK5mpUlEq1FrRdrruHURCgFQAAGZZJREFUH41NYXo2jApPEVY2F37DQeFe7sHboTA+vB491kXFBa1MGPDYJJczfKINB9U5BDNZrsFhv9gNU+wVzb1sp8dOk1fwugJyu7ZUDaQBwzxZLLIQO9Kdyu2yRn/PJeA5P+RHRNPRXO1FS406/dVkxIDHJrl0Wr44EsBcREd9hQdtderdELmsLo2lsKpWs1ldhRsbDqr4EAcMDygLN6JqR5UY5bzLqmiu0z0tOsR1pcBZY7JjwGOTWH51Dlvpn2uvVaoUVsglf0D031naWIGacjVKYYVc8yw+uXkbt+6Wwq5WtBTW6pyFwhrOD0d3WVV7iAO5X1uqvoJ3G76frS484u1G1JorGTHgsck9baUrugLIpXRYvVb2cfHXDlYT4aPX1VqFS2GtVh9duh5AKKyhtrwESxoq5mFkcsmlSst/ew7Xxu/usip2HyYGPNk/H204yB0eWTDgsUkuOTziIa7qCiD+2iH7Z2MdlhVchee6na7qKtzIao+n+CsaNXdZc+mFJe7Bjvpy1CnScFAwXhlWvtuH/nsHE9OzKClyYU2Lz76BkSUMeGxiNYfnuv8ORgMzKHK70Nmm5g1hdXUZ0fRYwrKKD/Fce6XEAmkF50qwmpsSO1xVsbwwIZccHpVznYw7PLns3q9u8cFbouYuq0wY8Ngk3lsm810hHkqrmqtQ7lGrFFZwW1xR/+vGFIKhCCo8RfhMkzqlsILVeQKipbCXYw0H1XyIA9Z7PMWq2RSr/BNy6YXVp/Cp3y7DE9NKj7V4l3N170GZMOCxidUzfPjawXpvC1VLYYVccp3ODvqh6cBCnxcLfepV/gmxE+YzzNlYYAbDk3fgckWvLRVZ3T3UDA0HVdwNyzWHh9/vcmHAYxerW+kKH+yYLNsWsepfHjnlhfG6AmAtN0W8olnZVIVKxRoOCla7eF8dn8bUTBjlniKsUqjhoJBLDs/MXASXRtQ91kVGDHhsYmXXYjYcwYXhuzeEog9xwNoqHGA1Wy69nVQ+MNTIys5Fn4KHYCazusMjXv2ta/WhuEi9x0cuOTznh/0IazoWVJWixafesS4yUu+KnSdW+lpcHAkgFNFQV+HB4vry+RmYhKzkD0zeDuHfd0thuxStZrP6mjRaCsuHOGBt50LlDsuC1R2e+Jl/al5XxjS67PmZ8R1pFSv/ZMSAxyZWHk6xL4+2GqVvCCsJ3qKKZklDhXKlsILbbS0RfuDWbdwMhu6WwqrZcFDItnMxF9Fwbkg0HFTzIQ5Yz6NT/bVyLjs8qu9Iy4gBj02slHnGbwg1vzyEeG5K+s/0KdrK3ozV5O41LIXN2gD0w+sBzIY1+MpKsFTBhoOClR5P/jtz+GhsGoC696HVHR5d12O5YSoH0rJhwGMTKxU1fIhHWQkOY31SFP7ysJrDo/oq3Cjb+WPGM6HcClb+CVbOaTt79x5srytHQ2XpfAxLOlZ3eIYn72B8ahbFbhfuW6RmfzUZMeCxSba8lFH/DEb8M3C7gM5WtQOebHOlJTQcVHeucm51wK30eG5Kmj+PNxxUOzjMNk8Ac52A+D0IZA4Oz8R2WauV32WVCQMem2R7OImk0pXN1ahQtBRWyJbD89HYNKZmo6WwKxVsOChYyXW6E4rgw+tTANTeDROy5fAwOIyyksMTP/Vb3evKZXGHx3hUCcmDAY9NRP+PrF+0Cq+WhGz5AyI4VLUUVrDy6u/c0CQimo6mapbCAsb7MPXPxqdmMXgr2nCwS8EmekbZ8uiiu6w89RuI34dWiixUT1eQjbpPD5tlezipXt5plC3fiTkpUdkScIHEc45UrvwTRI8ns4eTuK4+s6AKVd6S+RyWdDLNEwD8e2IagZkwvCVurFqo7i4rkD2XLtpwMFr5p/p3lmwY8NgkU15KKKzh3LC4IbgCiL3+S/PnKh9WaJRtngB2WE6W6T6MLzo4V9ny6MSZf+taa1Ci8C4rYLwPzSfrwrAfcxEdDZWlaK1V91gXGal95doo3j049c8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    "f(t) = mod1(t, 1.0)\n",
    "\n",
    "figure()\n",
    "ts = LinRange(-3, 3, 201)\n",
    "\n",
    "plot(ts, f.(ts))\n",
    "xlabel(L\"t\")\n",
    "ylabel(L\"f(t)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fourier_f (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Fourier_f(t, N)\n",
    "    \"\"\" Return the truncated Fourier series of the sawtooth function f(t)\n",
    "    \"\"\"\n",
    "    F = 0.5\n",
    "    \n",
    "    for ν=1:N\n",
    "        F -= sin(2π*ν*t)/(π*ν)\n",
    "    end\n",
    "    \n",
    "    return F\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   ],
   "source": [
    "fg, ax = subplots()\n",
    "\n",
    "@manipulate for N=slider(1:100, value=1, label=\"N\")\n",
    "    withfig(fg, clear=false) do\n",
    "        ts = LinRange(0, 2, 4001)\n",
    "\n",
    "        ax.cla()\n",
    "        ax.plot(ts, f.(ts), label=L\"f(t)\")\n",
    "        ax.plot(ts, broadcast(t -> Fourier_f(t, N), ts), label=\"N=$(N)\")\n",
    "        \n",
    "        ax.set_ylim(-0.2, 1.2)\n",
    "\n",
    "        ax.set_xlabel(L\"t\")\n",
    "        ax.set_ylabel(L\"f(t)\")\n",
    "        ax.set_title(\"Gibbs phenomenon for truncated Fourier series\")\n",
    "#         ax.legend()\n",
    "    end\n",
    "end"
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