2.1-FourierSeries-Symbolic.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "colonial-insider",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def L2_inner_product(f, g, tau):\n",
    "    # you can define t here to not rely on a global definition\n",
    "    t = symbols('t', real=True)\n",
    "    return (integrate(f*conjugate(g), (t, 0, tau))/tau).simplify()\n",
    "\n",
    "def L2_norm(f, tau):\n",
    "    return sqrt(L2_inner_product(f, f, tau))\n",
    "\n",
    "def L2_coefficient(f, g, tau):\n",
    "    return (L2_inner_product(f, g, tau)/L2_norm(g, tau)**2).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "closing-tenant",
   "metadata": {},
   "source": [
    "# Function to Approximate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "reserved-timothy",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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xuldL05FExEZe/HgXB/OKaRziz9NDNAz+c6jIiUiNe//bw3y5Lw9/H09mpibjrXV2ETnni70n+Ns3hwGYOiKJ0AAfw4nci36aikiNOnCiiMkrXOvsEwa0p0VkA8OJRMQu8ksqeHi+6yj2u7s1o2ebSMOJ3I+KnIjUmEqHk7HzMyitcHJTq0b88oZmpiOJiI08vWw7OQWlNI8I5NEB7UzHcUsqciJSY95Yd4DNmWcI9vdm6ogkrbOLSJWPt2azeHMWnh4wIzWFQF8Ng18JFTkRqRHbj+Uz6zPXOvszQzoSGxZgOJGI2EVuYSkTF28F4A+9W9KlWUPDidyXipyIVLuySgdpczOocFj079iYOzs3MR1JRGzCsiwmLtrK6ZIK2seE8Ke+Gga/GipyIlLtZq7aw+7jhTRq4MukOxO1zi4iVeZvPMpnO3Px9fLkpVHJ+HqrilwN3XoiUq2+P3SKOesOADD5zk5ENNA6u4i4HDlVwrPLdgCQ1q8N7RqHGE7k/lTkRKTaFJe51tktC0Z2aUq/jo1NRxIRm3A6LcbNz6CorJKuzRry2x4tTEeqE1TkRKTaTFqxk8xTJTQJC+DJwR1MxxERG3nn60N8e/AUgb5ezEhNxktHsVcLFTkRqRard+fywbeZAEwbmUSwv9bZRcRl7/FCXly5C4DHBrWnWUSQ4UR1h4qciFy1MyXlPLLAtc5+740JdG/ZyHAiEbGLinPD4OWVTnq1ieT/XRdvOlKdoiInIlftiY+2k1tYRsvIIMb3b2s6zhXTAz0i1e/V1fvYcjSf0AAfpo5I0lHs1UxFTkSuyrKMYyzLOIaXpwczU1Pw9/EyHUlEbGLL0TP85fN9ADw3NJHoEH/Dieoeo0VuypQpXHvttQQHBxMVFcXQoUPZvXv3j15nzZo1eHh4XHTatWtXLaUWkX85XlDKEx9tA+C+m1uRHBdmNpCI2EZphYMxc9NxOC1uT4phSHKs6Uh1ktEit3btWu677z7Wr1/PqlWrqKyspF+/fhQXF//kdXfv3k12dnbVqXXr1rWQWET+xbIsHlm4hTMlFXRqEsoDfVqZjiQiNjLtk93sP1FMZLAfz92RaDpOnWX0FWpXrlx5wdvvvPMOUVFRbNy4kZ49e/7odaOioggLC6vBdCLyYz78/ghrdp/A19uTmanJ+HjVpWdqWKYDiLi1b/af5K0vDwIwdXgSDYN8DSequ2z1kzc/Px+A8PDwn7xs586diYmJoW/fvqxevfoHL1dWVkZBQcEFJxG5OpknS3juH6519vG3taV1dLDhRCJiF4WlFYybnwHAf18Xz83togwnqttsU+QsyyItLY2bbrqJxMQfvgs2JiaGOXPmsHDhQhYtWkTbtm3p27cv69atu+Tlp0yZQmhoaNUpLi6upj4FkXrB4bQYOz+dknIH1yeEc++NCaYjiYiNPPePHWSdOUtceACPDWpvOk6dZ/Sh1X93//33s2XLFr788ssfvVzbtm1p2/b8vEG3bt04cuQI06dPv+TDsRMmTCAtLa3q7YKCApU5kavw1pcH+P7QaYJ8vZg+MhlPrbOLyDmf7TjOvA1H8fCAGSNTaOBnm5pRZ9niHrkHHniApUuXsnr1apo2bfqzr3/DDTewd+/eS77Pz8+PkJCQC04icmV25xQy/ZM9ADw5uANx4YGGE4mIXZwqLufRRVsB+F2PFlyX8NNPk5KrZ7QqW5bFAw88wOLFi1mzZg0JCVf2EM3mzZuJiYmp5nQi8u/KK52kzUun3OGkb7soUrvqnm0RcbEsi8cWbyWvqIw20Q0Yc2sb05HqDaNF7r777uODDz7go48+Ijg4mJycHABCQ0MJCAgAXA+NZmVl8d577wEwa9YsmjdvTseOHSkvL+f9999n4cKFLFy40NjnIVIf/OXzvWw/VkDDQB+mDO+kdXYRqfJR+jE+3paDt4bBa53RIjd79mwAevfufcH577zzDr/61a8AyM7OJjMzs+p95eXljBs3jqysLAICAujYsSPLly9n4MCBtRVbpN7ZlHmaV1e71tkn3dmJqGCts4uIS3b+WZ48Nwz+p76tSWwSajhR/WL8odWf8u67717w9vjx4xk/fnwNJRKR/3S23MG4eRk4LRiaEsvATnoag4i4WJbF+AVbKCitJDkujD/0bmk6Ur1ji4MdRMS+Xly5iwN5xTQO8eeZIVpnF5Hz3l9/mC/25uHv4xoG965Tw+DuQbe4iPygL/fm8e7XhwCYOiKJ0EAfs4FExDYO5hUzacVOAB7t346WkQ0MJ6qfVORE5JLyz1bw8ALXOvsvb2hGzzaRhhOJiF1UOpyMnZdOaYWTG1tFcHe35qYj1VsqciJySc8s2052finNIwKZMLCd6TgiYiNvrDvApswzBPt5M22EhsFNUpETkYus3JbNok1ZeHrAjNRkAn21zi4iLjuOFTDrM9cw+NNDOhIbFmA4Uf2mIiciFzhRWMbExa4pgdG9WtKlmdbZRcSlrNJB2rx0KhwWt3WMZtg1TUxHqvdU5ESkimVZTFy8lVPF5bRrHMyfbmltOpKI2MhLq/ayK6eQiCBfJt+pYXA7UJETkSoLNh5l1Y7j+Hh58NKoFPy8tc4uIi4bDp3ijXX7AZgyrBMRDfwMJxJQkRORc46eLuGZZTsAGHNrG9rHhBhOJCJ2UVxWydj5GVgWDL+mKf06NjYdSc5RkRMRnE6Lh+dvoaiski7NGvL7nlpnF5HzJq/YyeGTJcSG+vPUkA6m48i/UZETEd79+hDfHDhJgI8XM0Ym46UpARE5Z83uXP7vW9drnk8fmUyIv4bB7URFTqSe25dbxIsrdwEwcVB7mjcKMpzIHD1vW+RCZ0rKeWThFgB+1b053Vs1MpxI/pOKnEg9VuFwkjYvnbJKJz3bRHLX9fGmI4mIjTz50XaOF5TRIjKIR/prGNyOVORE6rHXVu9ny9F8Qvy9mTo8SVMCIlJlWcYxlmYcw8vTg5mpKQT46ih2O1KRE6mnth7N5y+f7wXguaGJNA71N5xIROwit6CUJz5yDYPf17slKXFhZgPJD1KRE6mHSiscjJmXTqXTYlCnGIYkx5qOJCI2YVkWjyzcwpmSCjrGhnB/Hw2D25mKnEg9NP2T3ezLLSIy2I/nhibqIVURqfLh90dYvfsEvt6evDQqBV9vVQU709+OSD2z/sBJ3vrqIAAvDu9EeJCv4UT2Y1mmE4iYkXmyhOf/4RoGf7hfW9pEBxtOJD9FRU6kHiksrWDcuXX2/7o2jj7tok1HEhGbcDgtxs3PoLjcwXUJ4dx7U4LpSHIZVORE6pHn/7GTo6fP0rRhAI/frnV2ETnvrS8P8N2hUwT5ahjcnajIidQTn+86ztwNR/DwgBkjk2ng5206kojYxO6cQqZ/sgeAJ27vQFx4oOFEcrlU5ETqgVPF5YxfsBWAX9+YwPUtIgwnEhG7KK90DYOXO5z0aRfFqGvjTEeSn0FFTqSOsyyLJ5ZsI6+ojNZRDRh3W1vTkUTERl75fC/bjxUQFujDC8M66Sh2N6MiJ1LHLc04xvKt2Xh7evDSqBT8fbTOLiIumzNP8+qa/QBMGtqJqBANg7sbFTmROiwnv5QnlrjW2R/s25rEJqGGE4mIXZwtdzB2XgYOp8UdKbEMSooxHUmugIqcSB1lWRbjF26hoLSS5Kah/LF3S9ORRMRGXly5iwN5xUSH+PHskETTceQKqciJ1FH/920m6/acwM/bkxmpKXh76dtdRFy+2pfHu18fAmDqiGRCA33MBpIrpp/sInXQobxiJi3fCcCjA9rRKqqB4UQiYhcFpRU8PD8DgLtuiKdXm0jDieRqqMiJ1DEOp0XavHTOVjjo1iKCe7o1Nx1JRGzkmaU7OJZfSrOIQCYObG86jlwlFTmROuaNdfvZlHmGYD9vpqcm46l1dhE555PtOSzcdBTPc8Pggb4aBnd3KnIidcjO7AJeWuVaZ39ycAeahAUYTiQidpFXVMbERa5h8N/3aknX5uGGE0l1UJETqSPKKh2MmZtOhcPi1g7RjOjS1HQkEbEJy7KYsGgrJ4vLadc4mIduaW06klQTFTmROmLWZ3vZlVNIeJAvU7TOLiL/ZuGmLFbtOI6PlwczU1Pw89YweF2hIidSB2w8fIo31rrW2Sff2YlGDfwMJxIRu8g6c5Znlm4HYMytbegQG2I4kVQnFTkRN1dSXsnYeRk4LRjWuQn9ExubjiQiNuF0Wjw8P4PCskquiQ/j9z01DF7XqMiJuLnJK3Zy6GQJMaH+PDWko+k4ImIjf/vmEF/vP0mAjxczUlPw0lHsdY6KnIgbW7vnBO+vzwRg2ohkQgO0zn419LxCqUv25Rbxwse7AJg4qD0JjYIMJ5KaoCIn4qbySyoYv8C1zn5Pt2bc1LqR4UQiYheVDidj56VTVumkR+tG3HV9vOlIUkNU5ETc1JNLt3G8oIwWjYJ4dIDW2UXkvNfW7CfjaD4h/t5MHZGke5vrMBU5ETe0fEs2H6Ufc62zpyYT4KspARFx2ZaVz8v/3AvAc0MTiQnVMHhdpiIn4mZyC0p5fIlrnf2PvVvROb6h4UQiYhelFa5h8EqnxcBOjRmSHGs6ktQwFTkRN2JZFo8u2srpkgo6xITwYF+ts4vIeTM+3c3e3CIaNfDj+aEaBq8PVORE3Mi8DUf4fFcuvl6evDQqBV9vfQuLiMv6Ayf565cHAXhxeCfCg3wNJ5LaoN8CIm7iyKkSnl22A4Cx/drQtnGw4UR1l2WZTiDy8xSVVTJufgaWBaO6xtG3fbTpSFJLVORE3IDTaTF2fgbF5Q6ubd6Q3/RoYTqSiNjI8//YwdHTZ2naMIDHb9dR7PWJipyIG3j7q4N8d/AUgb5ezBipdXYROe/zXcf58PsjeHjA9JHJBPtrGLw+UZETsbk9xwuZ+sluAB4f1IH4iEDDiUTELk4VlzN+geso9l/fmMANLSIMJ5LapiInYmMVDidp89Ipr3TSu20k/31dnOlIImITlmXxxJJt5BWV0TqqAeNua2s6khigIidiY3/5fB/bsgoIC/Rh6nCts4vIeUszjrF8azbenh7MTE3B30fD4PWRipyITaUfOcOrq/cB8NwdiUSF+BtOJCJ2kZNfyhNLtgHwQJ/WdGoaajiRmKIiJ2JDpRUO0ual43Ba3J4Uw2Cts4vIOZZlMX7hFgpKK0luGsofb25pOpIYZLTITZkyhWuvvZbg4GCioqIYOnQou3fv/snrrV27li5duuDv70+LFi14/fXXayGtSO15ceUuDpwoJirYj+eHJpqOIyI28n/fZrJuzwn8vD2ZkZqCj5fuk6nPjP7tr127lvvuu4/169ezatUqKisr6devH8XFxT94nYMHDzJw4EB69OjB5s2bmThxIg8++CALFy6sxeQiNefrfXm889UhAF4ckURYoNbZRcTlUF4xk5bvBOCR/u1oFdXAcCIxzdvkB1+5cuUFb7/zzjtERUWxceNGevbsecnrvP7668THxzNr1iwA2rdvz4YNG5g+fTrDhw+v6cgiNaqgtIKHF2wB4P9dH8/NbaMMJxIRu3CcGwY/W+GgW4sIftW9uelIYgO2uj82Pz8fgPDw8B+8zDfffEO/fv0uOO+2225jw4YNVFRUXHT5srIyCgoKLjiJ2NWzy3aQdeYs8eGBPDZQ6+wict6cdQfYePg0Dfy8mTYyCU8Ngws2KnKWZZGWlsZNN91EYuIPPycoJyeH6OgLX0MuOjqayspK8vLyLrr8lClTCA0NrTrFxWmHS+zp0+05LNh4FA8PmJGaTJCf0TvMRcRGdmYXMHOV6znkTw3uQNOGGgYXF9sUufvvv58tW7bw97///Scv+59bWta5V7i+1MbWhAkTyM/PrzodOXKkegKLVKO8ojImLHKts/+uZwuubf7D90qLSP1SVulgzNx0KhwWt7SPZkSXpqYjiY3Y4p/8DzzwAEuXLmXdunU0bfrjX6CNGzcmJyfngvNyc3Px9vYmIuLilybx8/PDz8+vWvOKVCfLsnhs8VZOFpfTNjqYtFvbmI4kIjby58/2siunkPAgX6YM66RhcLmA0XvkLMvi/vvvZ9GiRXz++eckJCT85HW6devGqlWrLjjv008/pWvXrvj46IWCxf0s3pzFJ9uP4+PlwcxRyfh5a51dRFw2Hj7F62v3AzD5zkQig3XHhFzIaJG77777eP/99/nggw8IDg4mJyeHnJwczp49W3WZCRMmcPfdd1e9PXr0aA4fPkxaWho7d+7k7bff5q233mLcuHEmPgWRq3LszFme+mg7AA/d0oaOsVpnFxGXkvJK0uZl4LTgzs5N6J8YYzqS2JDRIjd79mzy8/Pp3bs3MTExVae5c+dWXSY7O5vMzMyqtxMSElixYgVr1qwhJSWF5557jpdfflnTI+J2nE6L8Qu2UFhWSef4MH7fs4XpSCJiI1NW7OLwyRJiQv15ekhH03HEpow+R+5fByn8mHffffei83r16sWmTZtqIJFI7fnf9Yf5cl8e/j6ezBiZjLfW2UXknHV7TvC/6w8DMG1EMqEBeuqQXJp+c4gYcOBEEVM+dq2zTxzYnhaRWmcXEZf8kgrGnxsGv6dbM25q3chwIrEzFTmRWlbpcJI2L4PSCic9WjfiruubmY4kIjby1NJt5BSUktAoiEcHaBhcfpyKnEgte33tftKPnCHY35upI7TOLiLnrdiazZL0Y3ieGwYP8NVR7PLjVOREatG2rHxmfbYXgGfv6EhMaIDhRCJiF7mFpTy22DUM/sferbgmvqHhROIOVOREaklphYOx8zKodFr079iYoSlNTEcSEZuwLIsJC7dyuqSCDjEhPNi3telI4iZU5ERqyUur9rD7eCGNGvgy6c5ErbOLSJX5G47yz125+Hp5MnNUMr7e+vUsl0dfKSK14LuDp5jzxQEApgxLIqKB1tntzOKnp5FEqsuRUyU8s8w1DJ7Wrw3tGocYTiTuREVOpIYVlVUydn46lgUjuzTl1g7RpiOJiE04nRbj5mdQXO6ga7OG/LaHhsHl51GRE6lhk5bv5MipszQJC+DJwR1MxxERG3n7q4N8e/AUgb5ezEhNxktHscvPpCInUoNW78rl79+5XmJu2sgkgv21zi4iLnuPFzL1k90APDaoPc0iggwnEnekIidSQ04XlzN+oWud/X9ubE73llpnFxGXinPD4OWVTnq1ieT/XRdvOpK4KRU5kRryxEfbOFFYRsvIIB7p3850HBGxkVc+38fWrHxCA3yYOiJJR7HLFVORE6kBSzOO8Y8t2Xh5evDSqBT8fbTOLiIuGUfO8MrqfQA8NzSR6BB/w4nEnanIiVSz4wWlPLFkGwD339yKpKZhZgOJiG2UVjhIm5eOw2lxe1IMQ5JjTUcSN6ciJ1KNLMti/IIt5J+toFOTUO7v08p0JBGxkakrd7P/RDGRwX48d0ei6ThSB6jIiVSjD77LZO2eE/h6ezIzNRkfL32LiYjL1/vzePurgwBMHZ5EwyBfw4mkLtBvGZFqcvhkMZOW7wRg/G1taR0dbDiRiNhFYWkFD893HcX+39fFc3O7KMOJpK5QkROpBg6nxdh5GZSUO7g+IZx7b0wwHUlEbOTZZTvIOnOWuPAAHhvU3nQcqUNU5ESqwZtfHGDD4dM08PNm+shkPLXOLiLnrNpxnPkbj+LhATNGptDAz9t0JKlDVORErtKunAJmfroHgCdv70BceKDhRCJiFyeLypiwyPWQ6u96tOC6hHDDiaSuUZETuQrllU7S5mZQ7nByS/soRnZtajqSiNiEZVk8tngbeUXltIluwJhb25iOJHWQipzIVXj5n3vZkV1Aw0AfJg/rpHV2EamyJD2Lldtz8Pb0YGaqhsGlZqjIiVyhTZmneW2Na5198p2diArWOru7Uw+X6nLszFme/Gg7AA/d0prEJqGGE0ldpSIncgVKyisZOy8DpwV3dm7CgE4xpiOJiE04na5h8MLSSlLiwhjdq6XpSFKHqciJXIEXP97FwbxiGof48/SQjqbjiIiNvP/tYb7cl4e/j2sY3FvD4FKD9NUl8jN9uTePv31zGIBpI5MIDfAxnEhE7OLAiSImr3ANgz/avx0tIhsYTiR1nYqcyM+Qf7aChxdkAPDLG5rRo3Wk4UQiYheVDidj52dQWuHkxlYR3N2tuelIUg+oyIn8DM8s3U52finNIwKZMLCd6TgiYiNvrDvA5swzBPt7M22EhsGldqjIiVymlduyWbQ5C08PmJGaQqCv1tlFxGX7sXxmfeYaBn9mSEdiwwIMJ5L6QkVO5DKcKCxj4uJtAIzu1ZIuzRoaTiQidlFW6SBtbgYVDovbOkZzZ+cmpiNJPaIiJ/ITLMtiwqKtnCoup31MCA/donV2ETlv5qo97D5eSKMGvky+U8PgUrtU5ER+woKNR/ls53F8vDyYmZqMr7e+bUTE5ftDp5iz7gDgGgaPaOBnOJHUN/qNJPIjjp4u4ZllOwBIu7Ut7WNCDCeS2mBZphOIOygucw2DWxaM6NKUfh0bm44k9ZCKnMgPcDotxs3PoKiski7NGvK7ni1MRxIRG5m0YieZp0poEhbAk4M7mI4j9ZSKnMgPePfrQ6w/cIoAHy9mjEzGS1MCInLO6t25fPBtJuAaBg/x1zC4mKEiJ3IJ+3ILeXHlLgAeG9Se5o2CDCcSEbs4U1LOIwu2APA/Nzane8tGhhNJfaYiJ/IfKhxO0uZlUFbppGebSH5xfbzpSCJiI098tJ3cwjJaRgbxSH8Ng4tZV7xoWlFRQU5ODiUlJURGRhIeHl6duUSMeXX1PrYczSfE35upw5M0JSAiVZZlHGNZxjG8PD2YmZqCv4+X6UhSz/2se+SKiop444036N27N6GhoTRv3pwOHToQGRlJs2bN+O1vf8v3339fU1lFatyWo2d45fN9ADw3NJHGof6GE4mIXRwvKOWJj1zD4Pfd3IrkuDCzgUT4GUXupZdeonnz5rz55pv06dOHRYsWkZ6ezu7du/nmm2946qmnqKys5NZbb6V///7s3bu3JnOLVLvSCgdp8zKodFoMSophSHKs6UgiYhOWZfHIwi2cKakgsUkID/RpZTqSCPAzHlr9+uuvWb16NZ06dbrk+6+77jruvfdeZs+ezdtvv83atWtp3bp1tQUVqWnTPtnNvtwiIoP9eP6ORD2kKiJV/v7dEdbsPoGvtycvpabg46WnmIs9XHaRmz9//mVdzt/fnz/+8Y9XHEjEhG/2n+Ttrw4C8OLwTjQM8jWcSETsIvNkCc8vdw2DP9yvLa2jgw0nEjnviv5Jce+991JYWHjR+cXFxdx7771XHUqkNhWWVjBuvmud/b+ujaNPu2jTkUTEJhxOi7Hz0ykpd3BdQji/vinBdCSRC1xRkfvb3/7G2bNnLzr/7NmzvPfee1cdSqQ2PfePHWSdOUvThgE8frvW2UXkvL9+cYDvD50myNc1DO6pYXCxmZ81P1JQUIBlWViWRWFhIf7+54/oczgcrFixgqioqGoPKVJTPttxnHkbjuLhATNGJtPA74oXeUSkjtmdU8iMT/cA8OTgDsSFBxpOJHKxn/VbKywsDA8PDzw8PGjTps1F7/fw8OCZZ56ptnAiNelkURmPLnKts//mpgSubxFhOJGI2EV5pZMxc9Mpdzjp0y6K1K5xpiOJXNLPKnKrV6/Gsiz69OnDwoULLxgB9vX1pVmzZsTGarJB7M+yLB5fso28onJaRzVgbL+2piOJDXigh83E5S+f72VHdgENA314YXgnHcUutvWzilyvXr0AOHjwIPHx8frCFrf1UfoxPt6Wg7enBy+N0jq7iJy3KfM0r652DYNPurMTUcEaBhf7uuyDHTIzM6v+3KxZs58scVlZWVeeSqQGZeefrVpnf7BvaxKbhBpOJCJ2cbbcwbh5GTgtuCMlloGdYkxHEvlRl13krr32Wn7729/y3Xff/eBl8vPzefPNN0lMTGTRokXVElCkOlmWxfgFWygsrSQ5Low/9m5pOpKI2MiLK3dxIK+Y6BA/nh2SaDqOyE+67CK3c+dOQkND6d+/P9HR0QwaNIjf/va3PPDAA9x1111cc801REVF8e677zJt2jQeeOCBn/x/rlu3jsGDBxMbG4uHhwdLliz50cuvWbOm6mCLfz/t2rXrcj8NqefeX3+YL/bm4eftyczUZLy1zi4i53y5N493vz4EwNQRyYQG+pgNJHIZLvu32NGjR3nxxRc5duwYr7/+Om3atCEvL6/qNVV/8YtfsHHjRr766isGDBhwWf/P4uJikpOTeeWVV35W6N27d5OdnV110kuByeU4mFfMpBU7AXh0QDtaRjYwnEhE7CL/bAUPL8gA4K4b4unVJtJwIpHLc9kHO3Tu3JmcnBwiIyMZO3Ys33//PRERVzfXMGDAgMsuff8uKiqKsLCwq/rYUr9UOpykzUuntMJJ95YR3NOtuelIImIjzyzbTnZ+Kc0iApk4sL3pOCKX7bLvkQsLC+PAgQMAHDp0CKfTWWOhfkrnzp2JiYmhb9++rF69+kcvW1ZWRkFBwQUnqX/eWHeAzZlnCPbzZprW2UXk36zcls2iTVl4esDM1GQCfTUMLu7jsr9ahw8fTq9evYiJicHDw4OuXbvi5XXpyYZ/Fb7qFhMTw5w5c+jSpQtlZWX87//+L3379mXNmjX07NnzkteZMmWKRorruR3HCpj1mWud/akhHWkSFmA4kYjYxYnCMiYudh3F/vteLenSLPwnriFiL5dd5ObMmcOwYcPYt28fDz74IL/97W8JDg6uyWwXadu2LW3bnh9u7datG0eOHGH69Ok/WOQmTJhAWlpa1dsFBQXExWmhu74oq3SQNi+dCodFvw7RDL+mielIImITlmUxcfFWThWX065xMA/doudbi/v5Wfcf9+/fH4CNGzfypz/9qdaL3KXccMMNvP/++z/4fj8/P/z8/GoxkdjJS6v2siunkIggXyYP0zq7iJy3YONRVu04jo+Xaxjcz1vD4OJ+ruiJAO+8805157himzdvJiZGg41ysQ2HTvHGuv0ATB7WiUYNVOhFxOXo6RKeXbYDgDG3tqF9TIjhRCJXxugzOouKiti3b1/V2wcPHiQ9PZ3w8HDi4+OZMGECWVlZvPfeewDMmjWL5s2b07FjR8rLy3n//fdZuHAhCxcuNPUpiE0Vl1Uydn4GlgXDr2nKbR0bm44kbsQyHUBqlNNp8fD8LRSWVdKlWUN+31PD4OK+jBa5DRs2cPPNN1e9/a/nst1zzz28++67ZGdnX/DSYOXl5YwbN46srCwCAgLo2LEjy5cvZ+DAgbWeXext8oqdHD5ZQmyoP08N6WA6jojYyLtfH+KbAycJ8PFi+shkvHQUu7gxo0Wud+/eWNYP/9v33XffveDt8ePHM378+BpOJe5uze5c/u9b1z8Apo1MJsRf6+wi4rIvt4gXV7peDWjioPYkNAoynEjk6uj1iaROOVNSziMLtwDwq+7NubFVI8OJRMQuKs4Ng5dVOunZJpK7ro83HUnkqqnISZ3y5EfbOV5QRotGQTzSv53pOCJiI6+t3s+Wo/mE+HszdXiSjmKXOkFFTuqMf2w5xtKMY3h5ejBzVAoBvpoSEBGXrUfz+cvnrtcGf25oIo1D/Q0nEqkeKnJSJ+QWlPL4Etc6+x97tyQlLsxsIBGxjdIK1zB4pdNiUKcYhiTHmo4kUm1U5MTtWZbFIwu3cKakgo6xITzQR+vsInLejE93sze3iEYN/HhuaKIeUpU6RUVO3N7c74+wevcJfL09eWlUCr7e+rIWEZf1B07y1y8PAvDi8E6EB/kaTiRSvfQbT9xa5skSnvuHa5394X5taRNt/mXjRMQeisoqGXduGPy/ro2jb/to05FEqp2KnLgth9Ni3PwMissdXNc8nHtvSjAdSURs5Pl/7ODo6bM0bRjA47drGFzqJhU5cVtvf3mQ7w6dIshX6+wicqHPdx3nw++P4OEB00cm08DP6P69SI1RkRO3tOd4IdM+2Q3AE7d3ID4i0HAiqQv0HPi64VRxOeMXbAXgNzclcEOLCMOJRGqOipy4nfJKJ2PmplPucNKnXRSjro0zHUlEbMKyLB5fspW8ojJaRzVgbL+2piOJ1CgVOXE7r3y+l+3HCggL9OGFYZ00JSAiVZZmHGPF1hy8PT2YmZqCv4+GwaVuU5ETt5J+5AyvrtkPwKShnYgK0Tq7iLjk5JfyxLlh8Af7tqZT01DDiURqnoqcuI2z5a51dofTYkhyLIOSYkxHEhGbsCyLhxdkUFBaSXLTUP7Yu6XpSCK1QkVO3MaLK3dx4EQx0SF+PHtHR9NxRMRG3v82ky/25uHn7cmM1BS8vfTrTeoHfaWLW/h6Xx7vfn0IgBeHJxEWqHV2EXE5lFfM5OU7AXh0QDtaRTUwnEik9qjIie0VlFYwbn4GAL+4Pp7ebaMMJxIRu3A4LdLmpXO2wkG3FhHc06256UgitUpFTmzvmaU7OJZfSrOIQCYObG86jojYyBvr9rMp8wzBft5MT03GU8PgUs+oyImtfbI9h4WbjuLhATNGJhOkdXYROWdndgEvrdoDwJODO9AkLMBwIpHapyIntpVXVMbERa519t/3bEnX5uGGE4mIXZRVOhgzN50Kh8WtHaIZ0aWp6UgiRqjIiS1ZlsVji7dysricdo2DGXNra9ORRMRGZn22l105hYQH+TJFw+BSj6nIiS0t2pTFJ9uP4+PlwYzUZPy8tc4uIi4bD5/ijbWuYfDJd3aiUQM/w4lEzFGRE9vJOnOWp5duB+ChW9rQMVbr7FK7LMt0AvkhxWWVpM3LwGnBsGua0D+xselIIkapyImtOJ0W4xdkUFhWSef4MH7fs4XpSCJiI1M+3snhkyXEhPrz1GANg4uoyImtvPfNIb7adxJ/H09mjEzWOruIVFm75wTvr88EYNqIZEIDfAwnEjFPvyXFNvafKGLKx7sAmDiwPS0itc4uIi75JRWMX+AaBr+nWzNuat3IcCIRe1CRE1uodDhJm5dBWaWTHq0bcdf1zUxHEhEbeXLpNo4XlJHQKIhHB2gYXORfVOTEFmav2U/GkTME+3szdUSS1tlFpMryLdl8lH4MTw+YmZpMgK+OYhf5FxU5MW5bVj5//udeAJ69oyMxoVpnFxGX3IJSHl/iGgb/Y+9WdI5vaDiRiL2oyIlRpRUO0ualU+m0GJDYmKEpTUxHEhGbsCyLRxdt5XRJBR1iQniwr4bBRf6TipwYNXPVHvYcL6JRA1+eH5qodXYRqTJvwxE+35WLr5cnL41Kwddbv7JE/pO+K8SYbw+c5M0vDgAwZVgSEVpnF5Fzjpwq4dllOwAY268NbRsHG04kYk8qcmJEUVkl4xZkYFmQ2rUpt3aINh1JRGzC6bQYOz+D4nIH1zZvyG96aBhc5IeoyIkRk5bv4MipszQJC+CJ2zuYjiMCgB7Yt4e3vzrIdwdPEejrxYyRKXjpKHaRH6QiJ7Xu813H+ft3RwCYPjKZYH+ts4uIy57jhUz9ZDcAjw/qQHxEoOFEIvamIie16nRxOY8sdE0J3HtjAt1aRhhOJCJ2UeFwkjYvnfJKJ73bRvLf18WZjiRieypyUmssy+LxJds4UVhGq6gGjO/f1nQkEbGRv3y+j21ZBYQF+jB1eJKOYhe5DCpyUmuWZhxj+dZsvDw9mJmajL+P1tlFxCXjyBleXb0PgOfuSCQqxN9wIhH3oCIntSInv5QnP9oOwAN9WpHUNMxsIBGxjdIKB2PmpeNwWgxOjmVwcqzpSCJuQ0VOapxlWYxfuIX8sxUkNQ3lvptbmY4kIjby4spdHDhRTFSwH8/d0dF0HBG3oiInNe7/vs1k3Z4T+Hp7MjM1GR8vfdmJiMvX+/J456tDALw4IomwQF+zgUTcjH6jSo06lFfMpOU7AXikfztaRWmdXURcCkoreHjBFgD+3/Xx3Nw2ynAiEfejIic1xnFunf1shYMbWoTzP92bm44kIjby7LIdZJ05S3x4II8NbG86johbUpGTGjNn3QE2Hj5NAz9vpo1IxlPr7CJyzqfbc1iw8SgeHjAjNZkgP2/TkUTckoqc1Iid2QW8tGoPAE/e3oG4cK2zi4hLXlEZExa5hsF/17MF1zYPN5xIxH2pyEm1K6t0MGZuOuUOJ7e0j2Jk16amI4mITViWxWOLt3KyuJy20cGk3drGdCQRt6YiJ9Xuz5/tZVdOIeFBvkwZpnV2ETlv8eYsPtl+HB8vD2aOSsbPW8PgIldDRU6q1cbDp3l97X4AJg1NJDLYz3AiEbGLY2fO8tS5YfCHbmlDx9hQw4lE3J+KnFSbkvJKxs5Lx2nBnZ2bMKBTjOlIIlfEwjIdoc5xOi0eXpBBYVklnePD+H3PFqYjidQJKnJSbV74eBeHTpbQOMSfp4donV1Ezvvf9Yf5at9J/H08mTEyGW8Ng4tUC6PfSevWrWPw4MHExsbi4eHBkiVLfvI6a9eupUuXLvj7+9OiRQtef/31mg8qP2ndnhO8981hAKaNTCI0wMdwIhGxiwMnipjysWsYfOLA9rSIbGA4kUjdYbTIFRcXk5yczCuvvHJZlz948CADBw6kR48ebN68mYkTJ/Lggw+ycOHCGk4qPya/pILx59bZ7+7WjB6tIw0nEhG7qHQ4SZuXQWmFk5taNeKu65uZjiRSpxhdYBwwYAADBgy47Mu//vrrxMfHM2vWLADat2/Phg0bmD59OsOHD6+hlPJTnl62nZyCUhIaBfHogHam44iIjby+dj/pR84Q7O/N1BFJGgYXqWZu9SSFb775hn79+l1w3m233caGDRuoqKi45HXKysooKCi44CTV5+Ot2SzenIXnuXX2QF+ts4uIy7asfGZ9theAZ+/oSGxYgOFEInWPWxW5nJwcoqOjLzgvOjqayspK8vLyLnmdKVOmEBoaWnWKi4urjaj1Qm5hKY8t2QbAH3q35Jr4hoYTiYhdlFY4GDsvg0qnRf+OjRma0sR0JJE6ya2KHHDRuKxlWZc8/18mTJhAfn5+1enIkSM1nrE+sCyLiYu2cqq4nPYxIfypr9bZReS8l1btYffxQho18GXSnYkaBhepIW71OFjjxo3Jycm54Lzc3Fy8vb2JiIi45HX8/Pzw89MobXWbv+Eon+3MxdfLk5dGJePr7Xb/JhCRGvL9oVPM+eIAAFOGJRHRQD+DRWqKW/327datG6tWrbrgvE8//ZSuXbvi46O5i9py5FQJz/5jBwBp/drQrnGI4UQi1UN3Gl29orJK0ualY1kwsktTbu0Q/dNXEpErZrTIFRUVkZ6eTnp6OuCaF0lPTyczMxNwPSx69913V11+9OjRHD58mLS0NHbu3Mnbb7/NW2+9xbhx40zEr5ecTotx8zMoKquka7OG/LaH1tlF5LxJy3dy5NRZmoQF8OTgDqbjiNR5Rh9a3bBhAzfffHPV22lpaQDcc889vPvuu2RnZ1eVOoCEhARWrFjBmDFjePXVV4mNjeXll1/W9EgteufrQ3x78BSBvl7MSE3GS1MCInLO6t25/P0718/saSOTCPbXIyUiNc1okevdu3fVwQqX8u677150Xq9evdi0aVMNppIfsi+3kBdX7gLgsUHtaRYRZDiRiNjFmZJyHjk3DH7vjQl0b9nIcCKR+sGtniMn5lScW2cvr3TSq00k/++6eNORRMRGnvhoO7mFZbSMDGJ8/7am44jUGypyclleXb2PLUfzCQ3w4cXhSZoSEJEqyzKOsSzjGF6eHrw0KgV/Hy/TkUTqDRU5+Ulbjp7hL5/vA+C5oYk0DvU3nEhE7OJ4QSmPnxsGv+/mViQ1DTMbSKSeUZGTH1Va4SBtXgYOp8WgpBiGJMeajiQiNmFZFo8s3EL+2Qo6NQnlgT6tTEcSqXdU5ORHTftkN/tyi4gM9uP5OxJNxxERG/n7d0dYs/sEvt6ezExNxsdLv1JEapu+6+QHfbP/JG9/dRCAqcOTaBjkaziRiNjF4ZPFPL/cNQw+/ra2tI4ONpxIpH5SkZNLKiytYNz8DCwL/vu6OG5uF2U6kojYhOPcMHhJuYPrE8K598YE05FE6i0VObmk5/6xg6wzZ4kLD+CxQVpnF5Hz/vrFAb4/dJogXy+mj0zGU8PgIsaoyMlFPttxnHkbjuLhATNGptDAz+hutIjYyK6cAmZ8ugeApwZ3JC480HAikfpNRU4ucLKojEcXudbZf9ujBdclhBtOJCJ2UV7pJG1uBuUOJ33bRTGya1PTkUTqPRU5qWJZFo8v2UZeUTltohuQdmsb05FExEZe/udedmQX0DDQhynDO2kYXMQGVOSkykfpx/h4Ww7enh7MTNU6u4ictynzNK+tcQ2DT76zE1HBGgYXsQMVOQEgO/8sT3zkWmf/U9/WJDYJNZxIROzibLmDsfMycFowNCWWAZ1iTEcSkXNU5ASn0+Lh+VsoLK0kOS6MP/RuaTqSiFmW6QD28sLHOzmYV0zjEH+eGaJhcBE7UZET3v/2MF/uy8Pv3Dq7t9bZReScL/ae4G/fHAZg6ogkQgN9DCcSkX+n39j13MG8Yiav2AnAhAHtaBnZwHAiEbGL/LMVjF/gOor9lzc0o2ebSMOJROQ/qcjVY5UOJ2nz0imtcNK9ZQR3d2tuOpKI2MgzS7eTnV9K84hAJgxsZzqOiFyCilw99sa6A2zOPEOwnzfTtM4uIv9m5bZsFm3OwtMDZqSmEOirYXARO1KRq6e2H8tn1mfn1tmHdKRJWIDhRCJiFycKy5i42HUU++heLenSrKHhRCLyQ1Tk6qGySgdpczOocFj06xDN8GuamI4kIjZhWRYTFm3hVHE57WNCeOgWDYOL2JmKXD00c9Uedh8vJCLIl8nDtM4u8i8e6Hth/sajfLYzFx8vD2amJuPrrV8TInam79B65vtDp5iz7gAAU4Z1olEDP8OJRMQujp4u4dllOwBIu7Ut7WNCDCcSkZ+iIlePFJdVMnZeBpYFw69pSr+OjU1HEhGbcDotxs3PoKiski7NGvK7ni1MRxKRy6AiV49MWrGTzFMlxIb689SQDqbjiIiNvPP1IdYfOEWAjxczRibjpaPYRdyCilw9sXp3Lh98mwnA9JHJhPhrnV1EXPblFjJ15S4AHhvUnuaNggwnEpHLpSJXD5wpKeeRc+vsv+renO6tGhlOJCJ2UeFwkjYvg7JKJz3bRPKL6+NNRxKRn0FFrh544qPt5BaW0SIyiEf6a51dRM57dfU+thzNJ8Tfm6nDk3QUu4ibUZGr45ZlHGNZxjG8PD2YmZpCgK+X6UgiYhNbj+bzyuf7AHhuaCKNQ/0NJxKRn0tFrg47XlDKEx+51tnvu7kVKXFhZgOJiG2UVjgYMy+dSqfFoE4xDEmONR1JRK6AilwdZVkWjyzcwpmSChKbhPBAn1amI4mIjUz/ZDf7couIDPbjuaGJekhVxE2pyNVRH35/hDW7T+Dr7cnM1BR8vPRXLSIu3+w/yVtfHQTgxeGdCA/yNZxIRK6UfrvXQZknS3juH6519of7taVNdLDhRCJiF4WlFYyb7xoG/69r4+jTLtp0JBG5CipydYzj3Dp7SbmD6xLCufemBNORRMRGnv/HTrLOnKVpwwAev13D4CLuTkWujnnrywN8d+gUQb5aZxeRC3224zhzNxzBwwNmjEymgZ+36UgicpVU5OqQ3TmFTP9kDwBP3N6BuPBAw4lExC5OFZfz6KKtAPzmpgSubxFhOJGIVAcVuTqivNJJ2rx0yh1O+rSLYtS1caYjiYhNWJbF40u2kldURuuoBozt19Z0JBGpJipydcRfPt/L9mMFhAX68MKwTpoSEJEqSzOOsWJrDt6eHrw0KgV/Hw2Di9QVKnJ1wObM07y2Zj8Ak4Z2IipE6+wi4pKdf5YnlriGwf/UtzWJTUINJxKR6qQi5+bOljsYOy8Dh9NiSHIsg5JiTEcScXuW6QDVxLIsxi/YQkFpJclxYfyhd0vTkUSkmqnIubkXV+7iQF4x0SF+PHtHR9NxRMRG3v82ky/25uHn7cnM1GS8NQwuUufou9qNfbUvj3e/PgTA1BHJhAVqnV1EXA7lFTN5+U4AJgxoR8vIBoYTiUhNUJFzUwWlFTw8PwOAX1wfT682kYYTiYhdOJwWafPSOVvhoHvLCO7u1tx0JBGpISpybuqZpTs4ll9Ks4hAJg5sbzqOiNjIG+v2synzDMF+3kwbmYynhsFF6iwVOTe0clsOCzcdxfPcOnuQ1tlF5Jwdxwp4aZVrGPzpIR1pEhZgOJGI1CQVOTeTV1TGY4td6+y/69mSrs3DDScSqTvcfX6xrNJB2rx0KhwW/TpEM+yaJqYjiUgNU5FzI5ZlMWHRVk4Wl9OucTBjbm1tOpKI2Misz/ayK6eQiCBfJmsYXKReUJFzIws3ZbFqx3F8vDyYmZqCn7fW2UXEZePhU7yx1jUMPnlYJxo18DOcSERqg4qcmzh6uoRnlm4H4KFb2tAhNsRwIhGxi+KyStLmZeC0YPg1TbmtY2PTkUSklqjIuQGn0+Lh+VsoLKvkmvgwft+zhelIImIjUz7eyeGTJcSG+vPUkA6m44hILVKRcwN/++YQ3xw4SYCPFzNSU7TOLiJV1u45wfvrMwGYNjKZEH8fw4lEpDYZbwSvvfYaCQkJ+Pv706VLF7744osfvOyaNWvw8PC46LRr165aTFy79uUW8cLHrs9v4sB2JDQKMpxIROwiv6SC8Qtcw+C/6t6cG1s1MpxIRGqb0SI3d+5cHnroIR577DE2b95Mjx49GDBgAJmZmT96vd27d5OdnV11at26bh69WelwMnZeOmWVTnq0bsRdNzQzHUlEbOTJpds4XlBGi0ZBPNK/nek4ImKA0SI3c+ZMfv3rX/Ob3/yG9u3bM2vWLOLi4pg9e/aPXi8qKorGjRtXnby86ubRm6+t2U/G0XyC/b2ZOiJJUwIiUmX5lmw+Sj+Gl6cHM0elEOBbN38OisiPM1bkysvL2bhxI/369bvg/H79+vH111//6HU7d+5MTEwMffv2ZfXq1T962bKyMgoKCi44uYOtR/N5+Z97AXjujkRiQrXOLiIuuQWlPL7ENQz+x94tSYkLMxtIRIwxVuTy8vJwOBxER0dfcH50dDQ5OTmXvE5MTAxz5sxh4cKFLFq0iLZt29K3b1/WrVv3gx9nypQphIaGVp3i4uKq9fOoCaUVrnX2SqfFgMTG3JESazqSiNiEZVk8umgrp0sq6BgbwgN96uZTS0Tk8hh/kc7/fLjQsqwffAixbdu2tG3bturtbt26ceTIEaZPn07Pnj0veZ0JEyaQlpZW9XZBQYHty9yMT3ezN7eIRg38mHSn1tlF5Ly53x/h8125+Hp5MjM1BV9v48esiYhBxn4CNGrUCC8vr4vufcvNzb3oXrofc8MNN7B3794ffL+fnx8hISEXnOxs/YGT/PXLgwC8MKwT4UG+hhOJiF0cOVXCc//YAcC429rQtnGw4UQiYpqxIufr60uXLl1YtWrVBeevWrWK7t27X/b/Z/PmzcTExFR3PCOKyioZNz8Dy4JRXeO4pcPlF1oRqducToux8zMoLndwXfNwfn2ThsFFxPBDq2lpafzyl7+ka9eudOvWjTlz5pCZmcno0aMB18OiWVlZvPfeewDMmjWL5s2b07FjR8rLy3n//fdZuHAhCxcuNPlpVJvn/7GDo6fP0rRhAI/f3t50HBGxkbe+PMh3B08R6OvF9JHJeHnqKRciYrjIjRo1ipMnT/Lss8+SnZ1NYmIiK1asoFkz115adnb2BZty5eXljBs3jqysLAICAujYsSPLly9n4MCBpj6FavP5ruN8+P0RPDxg+shkgrXOLiLn7DleyLRPdwPw+KAOxEcEGk4kInbhYVmWZTpEbSooKCA0NJT8/HzbPF/uVHE5/V5aR15RGb++KYEnbtdrJYqY8NcvDvD88p3c2bkJL41KMR0HgPJKJ3e+9hXbjxXQu20k7/zqWh0AJSJVdLiTYZZl8cSSbeQVldEqqgEP39b2p68kIvXGK5/vZfuxAsICfZg6XMPgInIhFTnDlmYcY/nWbLw9PXgpNQV/H62zi4hL+pEzvLpmPwDPD00kKsTfcCIRsRsVOYNy8kt5Ysk2AB7o05pOTUMNJxIRuzhb7hoGdzgthiTHcnuShsFF5GIqcoZYlsX4hVsoKK0kqWkof7y5pelIInKOHZ46/OLKXRw4UUxUsB/P3tHRdBwRsSkVOUP+79tM1u05gZ+3JzNTk/Hx0l+FiLh8vS+Pd78+BMDUEUmEBWoYXEQuTe3BgEN5xUxavhOAR/q3o1WU1tlFxKWgtIJx8zMA+MX18fRuG2U4kYjYmYpcLXOcW2c/W+GgW4sIftW9uelIImIjzyzdwbH8UuLDA5k4UMPgIvLjVORq2Zx1B9h4+DQN/LyZNjIJT62zi8g5n2zPYeGmo3h4wIzUZIL8jG62i4gbUJGrRTuzC5i5yrXO/uTgDjRtqHV2EXHJKypj4qKtAPy+Z0uubR5uOJGIuAMVuVpSVulgzNx0KhwWt7SPZmSXpqYjiYhNWJbFxEVbOVlcTrvGwYy5tbXpSCLiJlTkasmfP9vLrpxCwoN8mTKsk9bZRaTKok1ZfLrjOD5eHsxITcbPW8PgInJ5VORqwcbDp3h9rWudffKdiUQG+xlOJCJ2kXXmLE8v3Q7AQ7e0oWOshsFF5PKpyNWwkvJKxs7LwGnBsM5N6J8YYzqSiNiE02nx8PwMCssq6Rwfxu97tjAdSUTcjIpcDZuyYheHTpYQE+rPU0O0zi4i5733zSG+3n+SAB8vZqam4K1hcBH5mfRTowat23OC/11/GHCts4cG+BhOJCJ2sS+3iCkf7wJgwsB2JDQKMpxIRNyRilwNyS+pYPyCLQDc3a0ZPVpHGk4kInZR6XAydn4GZZVOerRuxF3XNzMdSUTclIpcDXlq6TZyCkpJaBTEhAFaZxeR82av2U/GkTME+3szdYSGwUXkyqnI1YAVW7NZkn4Mz3Pr7AG+mhIQEZdtWfn8+Z97AXj2jo7EhAYYTiQi7kxFrprlFpby2GLXOvsfe7fimviGhhOJiF2UVjhIm5dOpdOif8fGDE1pYjqSiLg5FblqZFkWExZu5XRJBR1iQniwr9bZReS8mav2sOd4EY0a+DLpzkQNg4vIVVORq0bzNxzln7ty8fXyZOaoZHy9dfOKiMu3B07y5hcHAJgyLImIBhoGF5Grp6ZRTY6cKuGZZa519rH92tCucYjhRCJiF0VllYxbkIFlQWrXptzaIdp0JBGpI1TkqoHTaTFufgbF5Q6ubd6Q3/TQOruInDdp+Q6OnDpLk7AAnri9g+k4IlKHqMhVg7e/Osi3B08R6OvF9JHJeGlKQETOWb0rl79/dwSA6SOTCfbXMLiIVB8Vuau093ghUz/ZDcDjgzrQLELr7CLicrq4nPELXcPg996YQLeWEYYTiUhdoyJ3FSocTsbMS6e80kmvNpH893VxpiOJiI08/tE2ThSW0TIyiPH925qOIyJ1kIrcVXjl831syyogNMCHqSOSNCUgIlWWZhxj+ZZsvDw9eGlUCv4+GgYXkeqnIneFMo6c4ZXV+wB4fmgi0SH+hhOJiF0cLyjliSXbALj/5lYkNQ0zG0hE6iwVuSvwr3V2h9Pi9qQYBifHmo4kItXIuprrWhbjF2wh/2wFnZqEcn+fVtWWS0TkP6nIXYGpK3ez/0QxUcF+PHdHouk4ImIjH3yXydo9J/D19mRmajI+XvoxKyI1Rz9hfqav9+fx9lcHAXhxeBINg3wNJxIRuzh8sphJy3cCMP62trSODjacSETqOhW5n6GwtIKH57umBP77unhubhdlOJGIVKerOWDJ4bQYOy+DknIH1yeEc++NCdWYTETk0lTkfoZnl+0g68xZ4sMDeXxQe9NxRMRG3vziABsOnybo3DC4p4bBRaQWqMhdplU7jjN/41E8PFzr7EF+3qYjiYhN7MopYOanewB4anBH4sIDDScSkfpCRe4ynCwqY8Ii10Oqv+vRgusSwg0nEhG7KK90MmZuBuUOJ7e0j2Jk16amI4lIPaIi9xMsy+KxxdvIKyqnbXQwY25tYzqSiNjIn/+5h53ZBTQM9GHysE4aBheRWqUi9xOWpGexcnsO3p4ezEhN1jq7iFTZlHma2Wv2AzD5zk5EBWsYXERql4rcjzh25ixPfrQdgD/1bU1ik1DDiUTELkrKKxk7LwOnBUNTYhnQKcZ0JBGph1TkfoDT6VpnLyytJCUujD/0bmk6kojYyAsf7+JgXjGNQ/x5ZoiGwUXEDBW5H/D+t4f5cl8e/j6udXZvrbOLyDlf7D3Be98cBmDayCRCA30MJxKR+krt5BIOnChi8grXOvuEAe1pEdnAcCIRsYv8kvPD4L+8oRk9WkcaTiQi9ZmK3H+odDgZOz+D0gonN7aK4Jc3NDMdSURs5Oll28kpKKV5RCATBrYzHUdE6jkVuf/wxroDbM48Q7C/N9NGaJ1dRM5buS2bxZuz8PSAGakpBPpqGFxEzFKR+zfbj+Uz6zPXOvszQzoSGxZgOJGI2EVuYSkTFm0FYHSvlnRp1tBwIhERFbkqZZUO0uZmUOGwuK1jNHd2bmI6kojYhGVZTFy0ldMlFbSPCeGhWzQMLiL2oCJ3zsxVe9h9vJBGDXyZfKfW2UXkvPkbj/LZzlx8vVxHsft660eniNiDfhoB3x86xZx1BwDXOntEAz/DiUTELo6cKuHZZTsAGHNrG9rHhBhOJCJyXr0vcsVlrnV2y4IRXZrSr2Nj05FExCacTouHF2RQVFZJl2YN+V3PFqYjiYhcoN4XuUkrdpJ5qoQmYQE8ObiD6TgiYiPvfH2I9QdOEejrxczUZLx0FLuI2Ey9LnJrdufywbeZgGudPcRf6+wi4rIvt5AXV+4CYOLA9jSLCDKcSETkYvW2yJ0pKWf8Atc6+//c2JzuLRsZTiQidlHpsEibl0F5pZOebSL5xfXxpiOJiFyS8SL32muvkZCQgL+/P126dOGLL7740cuvXbuWLl264O/vT4sWLXj99dev6ONOWr6T3MIyWkYG8Uh/rbOLyHnLt2az5Wg+oQE+TB2epKPYRcS2jBa5uXPn8tBDD/HYY4+xefNmevTowYABA8jMzLzk5Q8ePMjAgQPp0aMHmzdvZuLEiTz44IMsXLjwZ3/sj7fl4OXpwczUFPx9vK72UxGROujZOzrSONTfdAwRkR9ktMjNnDmTX//61/zmN7+hffv2zJo1i7i4OGbPnn3Jy7/++uvEx8cza9Ys2rdvz29+8xvuvfdepk+fftkfM7egtOrP993ciuS4sKv9NESkjiitcFT9eVBSDEOSYw2mERH5acZeKLC8vJyNGzfy6KOPAq7l9MLCQnr16sW6desoKCi46DpffPEFvXr1uuB9PXr04K9//SsnT57Ex+figxXKysooKyur+hjj/74RgNYNPbmna9QlP46I1FPlJTjLSgAYf3M8hYWFhgOJSH0WHBz8k0/t8LAsy6qlPBc4duwYTZo04auvvqJ79+4UFBQQGhpqIoqIiIiI7eTn5xMS8uMj5MbukfuXfzXN4OBg8vPzmTZtGh9++CEbN2686LKdO3fmrrvuYuzYsVXnrV+/nttuu409e/YQHR190XX+/R45gOzsbK677jp27NhBkyZ6PdWrVVBQQFxcHEeOHPnJLza5PLpNq5duz+ql27N66fasXnXt9gwODv7Jyxgrco0aNcLLy4ucnBzAVehCQkIoKCggNjb2kn8BTZo04cyZMxe8r7i4GG9vb5o3b37Jh1Z/SHBwcJ34S7aLkJAQ3Z7VTLdp9dLtWb10e1Yv3Z7Vqz7dnsYOdvD19aVLly6sWrXqgvNXrVpF9+7dL3mdbt26XXT5Tz/9lK5du/6sEiciIiJSFxg9ajUtLY2//vWvvP322+zcuZMxY8aQmZnJ6NGjAZgwYQJ333131eVHjx7N4cOHSUtLY+fOnbz99tu89dZbjBs3ztSnICIiImKM0efIjRo1ipMnT/Lss8+SnZ1NYmIiK1asoFmzZoDr+Wz/vimXkJDAihUrGDNmDK+++iqxsbG8/PLLDB8+/LI/pp+f3wX/lavj5+fHU089pduzGuk2rV66PauXbs/qpduzetXH29PYUaum/Ovo2Ms5EkRERETEzoy/RJeIiIiIXBkVORERERE3pSInIiIi4qZU5ERERETcVL0rcm+++SYAkZGRdOnShS+++MJwIve1bt06Bg8eTGxsLB4eHixZssR0JLc1ZcoUrr32WoKDg4mKimLo0KHs3r3bdCy3NXv2bJKSkqpGQbt168bHH39sOladMWXKFDw8PHjooYdMR3FLTz/9NB4eHhecGjdubDqWW8vKyuKuu+4iIiKCwMBAUlJSLvkKUXVRvSpyc+fO5dFHHwXgyy+/pEePHgwYMOCCiRO5fMXFxSQnJ/PKK6+YjuL21q5dy3333cf69etZtWoVlZWV9OvXj+LiYtPR3FLTpk154YUX2LBhAxs2bKBPnz7ccccdbN++3XQ0t/f9998zZ84ckpKSTEdxax07diQ7O7vqtHXrVtOR3Nbp06e58cYb8fHx4eOPP2bHjh3MmDGDsLAw09FqRb2aH7n++utJTEzk7bffrpofad++PUOHDmXKlCmm47k1Dw8PFi9ezNChQ01HqRNOnDhBVFQUa9eupWfPnqbj1Anh4eFMmzaNX//616ajuK2ioiKuueYaXnvtNZ5//nlSUlKYNWuW6Vhu5+mnn2bJkiWkp6ebjlInPProo3z11Vf19hG2enOPXHl5ORs3bqRPnz4XnN+vXz++/vprQ6lELi0/Px9wlQ+5Og6Hgw8//JDi4mK6detmOo5bu++++xg0aBC33HKL6Shub+/evcTGxpKQkMB//dd/ceDAAdOR3NbSpUvp2rUrI0eOJCoqis6dO1c9jao+qDdFLi8vD4fDQVRU1AXnR0dHk5OTYyiVyMUsyyItLY2bbrqJxMRE03Hc1tatW2nQoAF+fn6MHj2axYsX06FDB9Ox3NaHH37Ipk2b9OhFNbj++ut57733+OSTT3jzzTfJycmhe/funDx50nQ0t3TgwAFmz55N69at+eSTTxg9ejQPPvgg7733nulotcLoS3SZ4OHhccHblmVddJ6ISffffz9btmzhyy+/NB3FrbVt25b09HTOnDnDwoULueeee1i7dq3K3BU4cuQIf/rTn/j000/x9/c3HcftDRgwoOrPnTp1olu3brRs2ZK//e1vpKWlGUzmnpxOJ127dmXy5MkAdO7cme3btzN79uwLXq+9rqo398g1atQILy8vjh8/fsH5ubm5REdHG0olcqEHHniApUuXsnr1apo2bWo6jlvz9fWlVatWdO3alSlTppCcnMyf//xn07Hc0saNG8nNzaVLly54e3vj7e3N2rVrefnll/H29sbhcJiO6NaCgoLo1KkTe/fuNR3FLcXExFz0D7T27dvXmwMZ6809cr6+vnTp0oXVq1dfcP6qVau44447DKUScbEsiwceeIDFixezZs0aEhISTEeqcyzLoqyszHQMt9S3b9+Ljqr8n//5H9q1a8cjjzyCl5eXoWR1Q1lZGTt37qRHjx6mo7ilG2+88aK5pj179tCsWTNDiWpXvSlyAGlpadx1110A7N69mw8++IDMzExGjx5tOJl7KioqYt++fVVvHzx4kPT0dMLDw4mPjzeYzP3cd999fPDBB3z00UcEBwdXPW8zNDSUgIAAw+ncz8SJExkwYABxcXEUFhby4YcfsmbNGlauXGk6mlsKDg6+6PmaQUFBRERE6HmcV2DcuHEMHjyY+Ph4cnNzef755ykoKOCee+4xHc0tjRkzhu7duzN58mRSU1P57rvvmDNnDnPmzDEdrXZY9cwrr7xixcXFWT4+PtY111xjrV271nQkt7V69WoLuOh0zz33mI7mdi51OwLWO++8YzqaW7r33nutZs2aWb6+vlZkZKTVt29f69NPPzUdq07p1auX9ac//cl0DLc0atQoKyYmxvLx8bFiY2OtYcOGWdu3bzcdy60tW7bMSkxMtPz8/Kx27dpZc+bMMR2p1tSrHTkRERGRuqTeHOwgIiIiUteoyImIiIi4KRU5ERERETelIiciIiLiplTkRERERNyUipyIiIiIm1KRExEREXFTKnIiIiIibkpFTkRERMRNqciJiJzTu3dvHnroIdMxREQum4qciIiIiJvSa62KiAC/+tWv+Nvf/nbBeQcPHqR58+ZmAomIXAYVORERID8/nwEDBpCYmMizzz4LQGRkJF5eXoaTiYj8MG/TAURE7CA0NBRfX18CAwNp3Lix6TgiIpdFz5ETERERcVMqciIiIiJuSkVOROQcX19fHA6H6RgiIpdNRU5E5JzmzZvz7bffcujQIfLy8nA6naYjiYj8KBU5EZFzxo0bh5eXFx06dCAyMpLMzEzTkUREfpTmR0RERETclO6RExEREXFTKnIiIiIibkpFTkRERMRNqciJiIiIuCkVORERERE3pSInIiIi4qZU5ERERETclIqciIiIiJtSkRMRERFxUypyIiIiIm5KRU5ERETETf1/pHXp3zLj5aMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f6568338eb0>"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = symbols('t', real=True)\n",
    "f = Piecewise(\n",
    "    (t, t < pi),   # t is zero before time pi\n",
    "    (t - pi, True),      # otherwise, t is 1\n",
    ")\n",
    "plot(f, (t, 0, 2*pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "stretch-miracle",
   "metadata": {},
   "source": [
    "# Fourier Series using $\\{e^{-ikt}\\}$, $k=[-\\infty, \\dots, \\infty]$ as Basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "utility-level",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} \\frac{\\left(-1\\right)^{k} - 1}{\\pi k^{2}} & \\text{for}\\: k > 0 \\vee k < 0 \\\\\\frac{\\pi}{2} & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((((-1)**k - 1)/(pi*k**2), (k > 0) | (k < 0)), (pi/2, True))"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = symbols('k', real=True, integer=True)\n",
    "tau = 2*pi\n",
    "w0 = 2*pi/tau\n",
    "psi_k = exp(-I*k*w0*t)\n",
    "\n",
    "ak = L2_coefficient(f, psi_k, tau)\n",
    "\n",
    "ak"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "17dc734a-9ee6-422e-9cdf-7b0e7a51c070",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\pi^{2}}{3}$"
      ],
      "text/plain": [
       "pi**2/3"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f*conjugate(f), (t, 0, 2*pi))/(2*pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "dfb5d6a5-a1ce-4e70-a711-abd7084e18da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.04348026911390171"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# creating ak list\n",
    "n = 3\n",
    "ak_list = []\n",
    "for k in range(-n, n+1):\n",
    "    if k == 0:\n",
    "        ak = np.pi/2\n",
    "    elif k % 2 == 0: # even\n",
    "        ak = 0\n",
    "    else:\n",
    "        ak = -2/(np.pi*k**2)\n",
    "    ak_list.append(ak)\n",
    "\n",
    "# compute error in approx p_n(t)\n",
    "norm_f_sq = np.pi**2/3\n",
    "error_norm_sq = norm_f_sq\n",
    "for ak in ak_list:\n",
    "    error_norm_sq = error_norm_sq - np.abs(ak)**2\n",
    "\n",
    "error_norm = np.sqrt(error_norm_sq)\n",
    "error_norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "29923832-2fbb-4981-bfb5-0874f6ca8332",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.0707355302630646,\n",
       " 0,\n",
       " -0.6366197723675814,\n",
       " 1.5707963267948966,\n",
       " -0.6366197723675814,\n",
       " 0,\n",
       " -0.0707355302630646]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ak_list"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}