Interactive Fourier series.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "19/11/2015\n",
    "\n",
    "**Jorge Vera Moreno**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Plotting a function defined by a series\n",
    "\n",
    "In this notebook, we will learn how to define, plot and manipulate a function defined by a series."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our function is the square wave, as defined by:\n",
    "\n",
    "$$f_\\text{square}(x) = \\frac{4}{\\pi} \\sum_{n = 1, 3, 5, \\ldots}^{\\infty} \\frac{1}{n} \\sin\\left(\\frac{n \\pi x}{L}\\right) \\qquad \\qquad $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we import **NumPy** and the modules _interactive_ and _display_, to allow interactivity in our plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\jorge\\Anaconda2\\lib\\site-packages\\IPython\\html.py:14: ShimWarning: The `IPython.html` package has been deprecated. You should import from `notebook` instead. `IPython.html.widgets` has moved to `ipywidgets`.\n",
      "  \"`IPython.html.widgets` has moved to `ipywidgets`.\", ShimWarning)\n"
     ]
    }
   ],
   "source": [
    "#Import NumPy\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "#Import functions to allow interactive plots\n",
    "from IPython.html.widgets import interactive \n",
    "from IPython.display import display\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "# The line below increases the default figure size. \n",
    "# Useful if you are on a large screen, you may want to\n",
    "# edit it to taste or comment it out\n",
    "# if you are on a small-screen laptop.\n",
    "plt.rcParams['figure.figsize'] = 9, 6 \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now set up our initial data: our X axis, and the period of the function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "L = np.pi # period of the function is [−L,L]\n",
    "x = np.linspace(0,5*np.pi,1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now define our function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def fourier_sum(xv,n):\n",
    "    '''function that creates the Fourier series for square wave.\n",
    "    Inputs:\n",
    "    x_values: array of x-values\n",
    "    n_terms: maximum value of n in summation from eqn(1) above\n",
    "    Returns:\n",
    "    fourier: array of y-values for Fourier square wave\n",
    "    '''\n",
    "    A=4/L #Amplitude multiplier\n",
    "    N=1.0 #First value for n in series\n",
    "    Y=0 #Initial value of sum\n",
    "    while N<=n: #While the current integer is not higher than the upper limit, do:\n",
    "        Y+=np.sin(N*xv)/N #Function definition\n",
    "        N+=2 #N can only take odd integers as values\n",
    "    return A*Y #Return the sum multiplies by the amplitude modifier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have defined our function, we will define another that will plot our square wave:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_fourier(Z):\n",
    "    \"\"\"function to draw plot based on current slider phase position\"\"\"\n",
    "    y = fourier_sum(x,Z) #Y-axis will be the square wave\n",
    "    plt.plot(x, y, linewidth=1, color='r')\n",
    "    plt.xlabel(r\"Angle\")\n",
    "    plt.ylabel(r\"Amplitude\")\n",
    "    plt.title(r\"Square wave\")\n",
    "    axes = plt.gca() #Set plot limits\n",
    "    axes.set_ylim([-1.5,1.5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot our function. Let's see what it looks like for $$n=3$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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tl81FL2AVV5HQVvGZzGzZYietdEcx1aRDB+v0+I9/eBdXukaMsAXBSkqy38chh9gyDZs3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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3dbf0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_fourier(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try for other values of $n$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2N6T8ocrvilz06cNkxnfMtnPDykz+YX+Z3PTqxSUN8s2SJdYVYYcdXEcSTT16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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7445390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_fourier(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There's no difference between $n=3$ and $n=4$, and that's to be expected, as the series is defined for odd integers. Let's try $n=5$, we should see a difference:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+OGamcPg1ZiZu6tWzRSkXLnTdEsqX35WZOHG0yGphhxlVK3cxzOSvfXvbImHt\nWtctoXzNncvKjFc6d948c4qiy+tp2XHWpYuTc6Kww0x5ua0r4cWu1XHHu9DCwW4m73TpkvnGrxRe\n7GbyjqOAX9hhhuNlvMW70OirrLS1kTp3dt2SwsAwUxjYzeQdhhkfcLyMtxhmoq+szDa1q1/fdUsK\nA8+JwsBuJu+wm8kHrMx4ixfu6FuwwDYrJW+wMhN9yWolb3y9wcqMD1iZ8RbDTPQtXMguJi8xzERf\nWRnQqpX/WwvERadOdk4EvCBvYYcZVma8xTATfQsXsjLjJZ4T0cdzwltNm1owXL480MMWfphhZcY7\nnTtzNlPULVjAyoyXkpWZAtgWJrYWLrSZmuQdByG/sMPMvHmszHiJd6HRx7tQbzVrZnvQrFjhuiWU\nq0WLeE54jWHGY7xwe6tTJ4aZqOMAYO8x5EcbPye8xzDjofXrgVWrbBoqeaNlS2DDBvt3pWjiAGDv\ncRBwtDHMeI9hxkNlZbYEf73C/REDJ8JxM1HHC7f3GGaijWNmvOdgrZnC/aRftMh2tSVvsaQeXZWV\nFvJ54fZWp04M+FHGMTPeY2XGQ4sW8aLtB4aZ6Fq6FGjRgutpeK1DB2DxYtetoFyxWuk9hhkPMcz4\ng91M0cWLtj86dmSYiSpVdjP5wcGmxAwzlB3unB1dXGPGHx062PWGomfFCqtUNmniuiWFJVmtDHD9\nJYYZyk6HDjbugqKHlRl/sJspunhO+KNJE1t/KcCZrwwzlJ327XnhjiqeE/5gN1N0Mcz4J+DPCoYZ\nyg4rM9FVVsYZfn5o3Rr4+Wdbg4miheNl/BPwZwXDDGWHlZnoWrzY/v+Rt+rVA9q1Y8iPIk7L9g8r\nMx5QZZjxS/v2wJIl3FgviliZ8U/HjhwEHEX8nPBPwGPJCjPMrFplq9U2beq6JYWnQQOgcWNg2TLX\nLaFsJVfFJu9xEHA0MeD7h91MHmDa9hfHzUQTw4x/OAg4mnhO+IfdTB5gmPEXx81ET2WlVdPatHHd\nksLEtWaiqazMxjuR99jN5AHuy+QvVmaip7zcdj0vLnbdksLEbqZoYmXGP+xm8gB/Qf3Fykz0cCaT\nv9jNFD2qNpmBlRl/sJvJA0uW8MLtJ1ZmoocDHf3FbqboWbkSqF/fJjSQ91iZ8QD7Qf3Vvj3DTNSw\nWukvdjNFDz8n/NWyJbB6NbB+fSCHK8www9Khv9jNFD0MM/5imIkenhP+CngxycINM/wl9Q+7maKH\nF25/Jbefiuj4AAAgAElEQVQ02LjRdUsoUzwn/BfgZ0XhhhlWZvzDykz08MLtr3r1LNCUl7tuCWWK\nN73+C/CzojDDDPtC/cXKTPQsXswBwH5r184+ICkaGPD9x8pMHrg4mP+aN7cdgteudd0SyhQv3P5r\n1w5YutR1KyhTvOn1X4BjyQovzFRUcHEwv4lwRlPUMMz4j5WZaOE54b+4dDOJyNEiMk1EfhCR69M8\nXyIiK0Tkm8SfG+v8phwvEwyOm4kWXrj9xzATLTwn/BdgN5Oz8oWIFAG4D8DhAOYD+EpEXlHVqdVe\n+oGqDsz4G7N0GAyOm4mO1aut+5W7yPuLYSZaOADYfzGpzOwDYIaqzlbVjQBGAjgxzeskq+/Kykww\nWJmJjuRFW7I7lShLDDPRwhtf/8VkAHAXAHNTvp6XeCyVAthfRCaKyOsi0rfO78q0HQxWZqKDM5mC\n0bYtw0xUVFXZYO22bV23pLC1bRvYoHiXo2Q1g9d8DaCbqq4RkWMAvASgT63vYGUmGG3bAgsXum4F\nZYJjA4LBykx0VFTYrMxttnHdksLWpo2FGVXfK8Muw8x8AN1Svu4Gq878j6quTPn7GyLygIi0VtWK\n6t9s6NCh9pexY1Gy334o8aHBlKJtW2DSJNetoEwwzASDYSY6eE4Eo1Ej28xz1SqgWbNaX1paWorS\n0tKcD+UyzIwH0FtEegJYAOBMAGenvkBEOgAoU1UVkX0ASLogA6SEmSlTgAMO8K3RlBBg+ZDytGQJ\ny+lBYJiJDlbwg5Ncf6mOMFNSUoKSkpL/fT1s2LCsDuMszKjqJhEZAuAtAEUAHlfVqSJyWeL5hwGc\nBuByEdkEYA2As+r8xvwlDQbDTHSUl/OcCELbttZ9UVVl2xtQeLEyE5zkZ8W22/p6GKcry6nqGwDe\nqPbYwyl/vx/A/Vl9U45QDwbDTHSUlwM77ui6FYWvfn2b/s4VyMOPYSY4AQ2ML7zbB1ZmgsEwEx2c\ntREcdjVFA296gxPQZ0VhhZmqKrsr4oXbfy1a2N5MGza4bgnVpbyclYKgMMxEw9KlDDNBYZjJQXK6\nHfdl8p+IfUCWl7tuCdVl6VKGmaAwzEQDA35wGGZywNJhsNjVFA3l5axWBoVhJhp4TgSHYSYHHC8T\nLIaZ8KusBJYvB1q1ct2SeEhOQ6VwY7UyOAwzOWCYCRbDTPgtX27rO7DrNRiszEQDu5mCE1DAL6ww\nw9JhsBhmwo/nRLAYZqKBM/yCw8pMDvgLGiyGmfDjHWiwGGbCb80a635t0sR1S+KBYSYHvHAHi7sE\nhx/HBgSLYSb8ktVKnzc+pITWrW3JlKoqXw/DMEO5Y2Um/NjNFCwG/PDj50Swiott3N6yZb4eprDC\nDO9Cg8UwE368cAerbVuuvRR2HI4QvAA+KworzPAuNFgMM+HHgB+sxo3tv2vWuG0H1YwBP3gMM1ni\nL2mwGGbCjwE/eFwZO9xYmQkew0yWeBcaLIaZ8GPAD16bNjwvwoznRPAYZrKwaROwciXQsqXrlsRH\n48aAKkvqYcaAHzyOmwk3VmaCF8DCeYUTZpYtsyBTVOS6JfEhwupM2LGbKXjsZgo3VmaCx8pMFvgL\n6gbDTLjxvAgew0y4sTITPIaZLPAX1A2GmfBSZZhxgWNmwo3nRPACWH+pcMIMf0HdYJgJr5UrgQYN\n7A8Fh2Nmwo1dr8FjZSYLDDNuMMyEF88JN9jNFG4cFB88hpks8BfUDYaZ8OI54QbDTHitX29/mjd3\n3ZJ4YZjJAkuHbjDMhBfPCTc4Zia8ystt40NuMhmsFi2A1auBjRt9O0RhhRnehQaPYSa8eE64wTEz\n4cWA70a9er5XLAsnzLCk7gbDTHjxnHCD3UzhxXPCHZ8/KwonzDBxuxHAyo6UI54TbrRoYati+1hS\npxzxnHDH5+nZhRVmmLiDx8pMePEu1A0RoFUrVmfCiOeEO+xmyhDDjBvJwY6qrltC1fGccIfjZsKJ\nlRl3GGYyVFHBC7cLyUXZVq503RKqjhdudzhuJpxYmXHHdZgRkSYicpOIPJr4ureIHO9bi3LVuDFQ\nv77rVsQTL9zhxAu3OzwnwokB3x3XYQbAkwA2ANg/8fUCALf51qJc8aLtDtfVCCd2M7nDcyKcGPDd\nCUGY2U5V74AFGqjqat9akw/+grrD8QHhxLtQd3hOhBPPCXdCEGbWi0ij5Bcish2A9b61KFf8BXWH\nJfXwWbMGqKqy7lcKHs+JcGJlxp0QhJmhAN4E0FVEngXwHoDrfWtRrvgL6g6nZ4dPsouJy7a7wTAT\nTqzMuOPzOVFc1wtU9W0R+RpA/8RDV6lq+D65GGbc4YU7fHjRdotjZsJn40bbH6hFC9ctiSdXYUZE\n+gFIXTxkAQAB0F1Euqvq1761Khe8cLvTti0webLrVlAqltPd4piZ8KmosMUM6xXOiiSR0qoVsHy5\ndX/78P+gtsrMXbAw0whAPwDfJh7fDcB4APt53pp88MLtDisz4cOZTG7xnAgfnhNuFRcDzZpZoGnd\n2vNvX2M8UtUSVR0Aq8jspar9VLUfgD0Tj4ULf0ndYUk9fNjN5BbDTPgsXcpzwjUfz4tMaj07quqk\n5Beq+h2AnXxpTT74S+oOS+rhw24mt1q3BpYts5I6hQMrM+45DjPfishjIlIiIgMSKwFP9KU1+eAv\nqTu8Cw0fXrjdSi2pUziwMuOe4zAzGMAUAL8BcFXi74N9aU0+eOF2h1Ozw4fdTO4x5IcLA757Pp4T\nmUzNXgvg7sSf8OIvqTvJhdnWrOEibWHBbib3khfu3r1dt4QAOyc6dHDdinhzGWZEZFaah1VVe/nQ\nntzxQ9StZHWme3fXLSGAd6FhwIHx4VJeDvTt67oV8eYyzAD4RcrfGwI4DQCvkrSl5C8pw0w4sJvJ\nPQ6MDxeOmXGvTRtg0qS6X5eDOsfMqOrSlD/zVPUfAI7zpTUUXbwLDRd2M7nHMTPhwmqle467mVJX\nAq4HYG8ARb60hqKLd6HhsWEDsHYtl213jWEmXBjw3XPczZRcCRgANgGYDeAMX1pD0cULd3hUVNg6\nJ9xk0q02bYAJE1y3gpLY9eqe4zBzkarOTH1ARLb1pTUUXZyeHR68Aw0HVivDo7ISWLHC9gcidxyv\nMzMqw8eyJiJHi8g0EflBRK6v4TX/TDw/UUT29OK45ANWZsKDYwPCgedEeCxbBjRvDhRxhIRTLioz\nIrITgL4AWorIKbAdsxVAc9ispryISBGA+wAcDmA+gK9E5BVVnZrymmMBbK+qvUVkXwAPAuif77HJ\nB23bAl984boVBHDWRlhwUHx4MOCHQ+PGtsXH2rVAo0aefuvaupl2AHACgBaJ/yatBHCpB8feB8AM\nVZ0NACIyEsCJAKamvGYggBEAoKpfiEhLEemgqos9OD55iXeh4cELdziwmyk8OF4mHEQ2f1Z07erp\nt64xzKjqSwBeEpH9VPUzT49qugCYm/L1PAD7ZvCargAYZsKGd6HhwTATDsmLtioHY7vGcyI8gg4z\nInK9qt4B4BwROafa06qqV+V5bK37JdaUTN43dOjQ//29pKQEJSUlOTWKcsS70PBYuhTo2NF1K6hh\nQ9twcvVqoGlT162JN4aZ8NhuO9v6pprS0lKUlpbm/G1r62aakvjvf9M8l2kQqc18AN1Svu4Gq7zU\n9pquice2khpmyAF2M4VHeTmw886uW0HA5ll+DDNuMcyEx5gxaR+uXoQYNmxYVt+2tm6mVxP/fSqr\n75i58QB6i0hPAAsAnAng7GqveQXAEAAjRaQ/gOUcLxNSzZoB69fbnwYNXLcm3njhDo9kyO/Z03VL\n4o3LFRS82rqZXq3lfaqqA/M5sKpuEpEhAN6CrSj8uKpOFZHLEs8/rKqvi8ixIjIDwGoAg/M5Jvko\ndWBX586uWxNvnM0UHqxYhkN5OdCjh+tWkI9q62a6q5bnvOhmgqq+AeCNao89XO3rIV4ciwKQLKkz\nzLjFykx4cDHJcOBspoJXWzdTafLvItIAwI4AqgBMV9UN/jeNIod3oeHAMBMePCfCgedEwctko8nj\nADwEILmlQS8RuUxVX/e1ZRQ9nJ7tHpdtDxeeE+HAMFPwMtmb6W4AA1R1BgCIyHYAXk/8IdqM07Pd\nSy7bXpzJqU2+a9sWmD7ddSuIYabgZbI308/JIJMwE8DPPrWHoowldfd40Q4XnhPuqfK8iIFMbt/+\nKyKvA3gh8fXpAMYn9muCqr7oV+MoYtq2BebOrft15B/OZAoXdjO5t2qVVSob5r2lIIVYJpWZhgDK\nAByS+LMk8dgJ2HLPJoo73oW6xzvQcGHXq3s8J2KhzsqMql4YQDuoEHAaqnu8cIcLA757nJYdC5nM\nZuoF4EoAPVNen/eieVSAeOF2j91M4cKA7x4DfixkMmbmJQCPAXgVts4M4NGieVRgeOF2jxfucGnS\nBNi0CVi3jmM2XOE5EQuZhJm1qvpP31tC0cfKjHvl5cC227puBSWlbvPRpYvr1sQTw0wsZBJm/iki\nNwN4G8D65IOq+rVvraJoatECWL0a2LgRqF/fdWviid1M4ZOsWDLMuMFNJmMhkzCzC4BBAA7F5m4m\nABjgS4souurVs5VnKyqADh1ctyaeeBcaPqxYulVeDmy/vetWkM8yCTNnAOjF/ZgoI8kLN8OMGwwz\n4cO1ZtwqLwf693fdCvJZJuvMTALAjV4oMxwE7Ba7mcKHa824xYAfC5lUZloBmCYiX2HzmBlOzab0\nWFJ3R9W6+HjhDheeE24xzMRCJmHm5sR/FYAAOBjAWb61iKKNlRl3Vqyw6b/bbOO6JZSqTRtgzhzX\nrYgvhplYqLObSVVLYRtLHg9gBGwg8IP+Nosii3eh7nCl03BiN5NbDDOxUGNlRkR2AHA2rApTDuB5\nAKKqJcE0jSKpTRugrMx1K+KJF+1wYsB3Z8MGYO1aoHlz1y0hn9VWmZkKm359gqoekFg4rzKYZlFk\n8S7UHa6nEU6czeROeTnQurUtXkgFrbYwcwqARQDeF5FHReRw2JgZoprxLtQdVmbCiQHfHXa9xkaN\nYUZVX1LVMwHsCKAUwNUA2onIgyJyZEDto6jhAGB3li4F2rVz3QqqjgHfHQb82MhkAPAqVX1GVY8H\n0A3ANwBu8L1lFE28cLvDNWbCKXWbDwoWw0xsZLJo3v+oaoWqPqKqh/rVIIo4VmbcYZgJp9RtPihY\nDDOxkVWYIapTq1a23kklx4oHjuMDwosVSzc4KD42GGbIW0VFNg1y2TLXLYkfVmbCixVLN1iZiQ2G\nGfIeZ2+4wTATXqzMuMEwExsMM+Q9XrjdYJgJL6414wa7XmODYYa8x5J68FR5FxpmrFa6wXMiNhhm\nyHuszARvxQqgcWNuMhlWPCfcYJiJDYYZ8h4rM8FjF1O4sZvJDYaZ2GCYIe/xLjR4DDPhxm6m4FVV\n2azK1q1dt4QCwDBD3uNdaPAYZsKNAT94y5cDTZsCxcWuW0IBYJgh7/EuNHgMM+HGrtfgcSZTrDDM\nkPd4Fxo8hplw4zkRPI6XiRWGGfIe70KDxzATbq1aWbcHt/kIDsNMrDDMkPd4Fxo8hplwKy62bT6W\nL3fdkvhgmIkVhhnyXps2tkOwquuWxAfDTPgx5AeLm0zGCsMMea9+fVvAbcUK1y2JD164w49hJlhL\nlgDt2rluBQWEYYb8wenZwWJlJvw4lixYDDOxwjBD/uD07GBxGmr4sTITLIaZWGGYIX/wwh2cykqu\ndBoFrFYGi2EmVhhmyB8sqQdn+XKbKcOVTsON1cpgLV3KMBMjDDPkD1ZmgsPxMtHAcyJYrMzECsMM\n+YOVmeAwzEQDu5mCs3EjsGoV0LKl65ZQQBhmyB+8Cw0Ow0w0sJspOMmlCurxIy4u+H+a/MG70OAw\nzERD27bW9UH+W7KE50TMOBkxKCKtATwPoAeA2QDOUNWt1vkWkdkAfgZQCWCjqu4TYDMpH7wLDQ7D\nTDS0b88wExSOl4kdV5WZGwCMU9U+AN5NfJ2OAihR1T0ZZCKG3UzBYZiJhjZtuNlkUBhmYsdVmBkI\nYETi7yMAnFTLa8X/5pDnOAA4OAwz0VBUZANSGfL9xzATO67CTAdVXZz4+2IAHWp4nQJ4R0TGi8il\nwTSNPJGszHCzSf8xzERH+/ZAWZnrVhQ+hpnY8W3MjIiMA9AxzVN/Sv1CVVVEavrEO0BVF4pIOwDj\nRGSaqn7kdVvJBw0b2oaTq1YBzZq5bk1hY5iJDoaZYCxdCuy8s+tWUIB8CzOqekRNz4nIYhHpqKqL\nRKQTgLRnt6ouTPx3iYiMAbAPgLRhZujQof/7e0lJCUpKSnJvPHkjWZ1hmPEXZ25ER7t2HAQcBFZm\nIqe0tBSlpaU5v9/V+uevALgAwB2J/75U/QUi0hhAkaquFJEmAI4EMKymb5gaZigkktOze/Z03ZLC\nVlYGdKipp5ZChZWZYDDMRE71IsSwYTV+3KflaszM7QCOEJHvARya+Boi0llExiZe0xHARyIyAcAX\nAF5T1bedtJZyw+nZ/lu7FtiwwfZmovBjmAkGq5Wx46Qyo6oVAA5P8/gCAMcl/j4TwB4BN428xBlN\n/isrsw9I4aS/SGjXDpg40XUrCh8rM7HDFYDJP7wL9d/ixfbvTNHAc8J/VVVARYV1c1NsMMyQf9q3\ntw9b8g/Hy0RLu3YMM36rqLBu1/r1XbeEAsQwQ/7p0IFhxm/JbiaKBm5p4D92McUSwwz5h2HGf+xm\nihZ2M/lv6VKGmRhimCH/MMz4j91M0dKypS0kuWGD65YULlZmYolhhvzToQPvQv3GbqZoqVePs/z8\nxmnZscQwQ/5JltS5P5N/2M0UPexq8teiRUDHdDvpUCFjmCH/NGxof5Yvd92SwsVupujhlgb+WryY\n50QMMcyQvzhuxl+szEQPKzP+YpiJJYYZ8hfDjH8qK21NDY4PiBauNeMvhplYYpghfzHM+KeiAmjR\ngouDRQ3XmvEXw0wsMcyQv1hS9w+7mKKJ54S/OAA4lhhmyF+szPiHg3+jid1M/lmzBti4kbvIxxDD\nDPmLYcY/XGMmmnhO+CfZxcRd5GOHYYb8xQu3f9jNFE0dO1pXCHmP42Vii2GG/MUw4x92M0VTMsxw\nMUnvMczEFsMM+Ythxj/sZoqmRo3sz7JlrltSeDj4N7YYZshfnLnhH3YzRRe7mvzBykxsMcyQv5o2\ntXL6qlWuW1J4WJmJrk6dgIULXbei8DDMxBbDDPlLhF1Nflm4EOjc2XUrKBeszPiDYSa2GGbIfwwz\n3quqsjDD8QHRxMqMPxhmYothhvzHMOO98nKgWTPblZyih5UZfyxaxDATUwwz5D/ehXpv4UL7d6Vo\n4jnhPVV2vcYYwwz5r3NnYMEC160oLAsW8KIdZazMeG/lSvtvs2Zu20FOMMyQ/7p0AebPd92KwsI7\n0GhjZcZ78+fbtYZbGcQSwwz5j5UZ7y1YwG6mKGNlxnusVsYawwz5j2HGe6zMRFvr1rb20rp1rltS\nOBhmYo1hhvzXpQvDjNd44Y62evU4y89rCxbYtYZiiWGG/NemDbB6NbB2reuWFA52M0Ufx814a/58\nBvwYY5gh/4nwwu01djNFHwfGe4uVmVhjmKFgcNyMd1S5O3Ah6NYNmDvXdSsKB7teY41hhoLRuTPv\nQr1SXm4beHL132jr2hWYN891KwoHu5lijWGGgsFBwN7hHWhh6NqVlRmvVFVZtZLnRWwxzFAw2M3k\nHQ7+LQzdurEy45XkXmUNGrhuCTnCMEPBYDeTd+bMAbp3d90Kyhe7mbzDLqbYY5ihYLCbyTsMM4Wh\nSxeblVZZ6bol0ceZTLHHMEPB6NKF4wO8MmcO0KOH61ZQvrbZxlYC5sJ5+Zs7l2Em5hhmKBjdu1tJ\nvarKdUuij5WZwsHp2d746ScG/JhjmKFgNGxod6F+LZw3bx5w/fXAxRcD777rzzHqMm0a8JvfAL/6\nFTB+vH/HYZgpHH6Om1m7FrjjDuCCC4B//QvYtMmf49SmogL485+BwYOBF1/07zg//QT07Onf96fQ\nY5ih4PToYRcdr339NbDPPjb2YM89gUsuAa67zhaXC8ro0cBBB9nWDdtvDxx7LPDUU94fp7LSBjt2\n7er996bg+TWjaelSYL/9gC+/BA48EHjoIeCEE4A1a7w/Vk2++w7YZRfrRuvfH/jTn4Crr/bnvGRl\nJvaKXTeAYqRHD2D2bGD//b37nhUVwMknA//4B3DGGfbYuecCAwbYCrm//a13x6rJZ58Bl18OjBsH\n7LGHPXbCCcDBBwO9ewMHHODdsRYtsgoXp6AWBj/WmqmsBM46Czj8cODvf7ftRAYPBgYNsqD/zDP2\nmJ/KyoBjjgHuvBM45xx77Mwz7bx86CE7X7w0ezbDTMyxMkPB8aMy8/vfAyeeuDnIAECrVsBLLwF/\n+xswaZK3x6tuzRrg/POBRx7ZHGQAYIcd7KJ96aXAxo3eHY9dTIXFj26mJ5/c3MWUDC3FxcATTwCT\nJ1uXk59Ugcsus5uKZJABgJYtgRdeAG680dsAt2GDhScOAI41hhkKTs+e3oaZSZOA114D/vKX9Mf6\ny19sDIuf3U233w706wecdNLWz510knUjPPywd8djmCksyWqlV1atAm66CfjnP4Gioi2fa9QIeOwx\nG1u2fLl3x6zurbeAKVOAYcO2fq53b6vK3Hijd8ebN8/WmClmR0OcMcxQcLyuzNx6q12YW7RI//wl\nl9iA4zff9O6YqRYvBu6/3wJNOiLAbbcB//d/3lVnGGYKS69ewMyZ3n2/Rx+1MTL9+qV//he/sPFc\nf/+7d8dMVVlp5+Ttt9fcFXrddcDYsd793BwvQ2CYoSB5eRc6ezbwzjsWWGpSXGxdTddf78+U8L/+\n1Urptc2i2HtvuxsdOdKbYzLMFJaOHYHVq4GVK/P/Xps2AcOH2+97bW66ybpAKyryP2Z1zz1nm6Cm\nq1QmNW9u5+0993hzTI6XITDMUJB69LAPYy+6fR5+2KacNm9e++tOPNHK7V5XZ8rKgKefthkadbnq\nKhtT4wWGmcIiAmy7LTBrVv7f6403bAzO3nvX/rqePW3Q/PDh+R8zlapVIf/857oHGA8ZYufP6tX5\nH5eVGQLDDAUpuRHc0qX5fZ/KShvEeNFFdb9WBPjd72xWhZcefhg47TSgQ4e6X3vsscD339uffM2a\nxfU0Ck2vXsCPP+b/fZ54IrNzAgD++EfggQdsjI1Xxo2zQHPkkXW/tmtXmzruxdozDDMER2FGRE4X\nkckiUikie9XyuqNFZJqI/CAiddROKRJ69sy/q+ndd608v8sumb3+jDOAGTOA//43v+MmrV9vHwRX\nX53Z6+vXt2mxTz6Z33FV7UNvu+3y+z4ULtttl//4kbIy4P33t5zVV5tevYBDDrHqiFfuusuWQsh0\n2veFFwIjRuR/3NmzGfDJWWVmEoCTAXxY0wtEpAjAfQCOBtAXwNkislMwzSPf9OyZf0n9qaesiylT\n9evbrCavqjMjRwK77grsvHPm7xk82D448tlUcNEiG49QV9caRYsXg4Cfeca6VLP53bj6apv15MV4\nsu++s9mFqVOx6zJwIPDNN9Z1mg9WKwmOwoyqTlPVumru+wCYoaqzVXUjgJEATvS/deSr3r2BH37I\n/f2rV9tMiLPPzu59l14KvP12/hdOVRtrcM012b1v552Btm2BTz/N/dgzZrAqU4i86Gb6z3+yCxKA\nrVjdqJFNpc7X3XcDV1yR3WKODRtaV+1zz+V+3HXrLOQzzMRemMfMdAGQurLSvMRjFGV9+uQXZt56\ny6aXtm2b3fuaN7dqzv33535sAPjgA+tmOuqo7N972mnAqFG5H3vGDNsqgQpLvt1MCxbYvmADBmT3\nPhGrzuQ7q2jRImDMGNuTLFunnZbfuJmZM228DNeYiT3fwoyIjBORSWn+nJDhtwhwYx0KTJ8++Q2E\nHTPGZmLk4sorgccfz2/Q4/Dh9gFQL4dT59RT7cKda1n/xx8ZZgpRz55WMcy1C/Kll2yQ+TbbZP/e\ns84CJkwApk7N7diATfM+80zblyxbJSX2e53risDff2/XFIo93+Ksqh6R57eYD6BbytfdYNWZtIYO\nHfq/v5eUlKCkpCTPw5Mv8ulm2rjRuphqWqSuLttua4MeR4ywkni2fvjB9mHKtSzet6+NefnyS9t4\nL1szZtieT1RYGja0SuO8ebnNynnxReDXv87t2A0a2Iq8w4fntnzA+vUWZt57L7fj169vv9NjxtgS\nBtn64Qe7plDklZaWorS0NOf3iwa5s3D1g4u8D+B3qrrVNBMRKQYwHcBhABYA+BLA2aq61S2EiKjL\nn4OyoGor9v70k+2hlI1x42zBr88/z/34H39s01enTcu+ujJkiO0vc+utuR//ppusnz+XFVj33NOm\nhO+zT+7Hp3AaMMCmSx+R5T1gRYWF9AULgCZNcjv2kiVW3Zg2LbOlBlKNGGHhPp91nF591WZC5fJB\n9stfAnvtlVsXF4WaiEBVM94R1dXU7JNFZC6A/gDGisgbicc7i8hYAFDVTQCGAHgLwBQAz6cLMhQx\nIrlXZ158MfcupqQDDrAwNXZsdu+rqACefTa3ik6qU08FRo/OfuHAykpg+nSr7lDh6dvX9jPK1quv\nAocdlnuQAYB27ay76YEHsnufqu1Wn+kSBTU54ghg4kSbXp6t779nZYYAuJvNNEZVu6lqI1XtqKrH\nJB5foKrHpbzuDVXdQVW3V9W/uWgr+WDHHbPvo6+qAl5+Of8wI2IzkbJd/fSRR6wc3qlTfsfffXdr\nw8SJ2b1v9mygfXvrpqLCs/POuYWZF18ETjkl/+Nfcw3w4IO2C3ymPvjAqoyZLJJXm4YNgaOPtvM7\nW1On2vWEYi/Ms5moUO26q61JkY0vvgBat/ZmsN/pp9sd3YQJmb1+zRrg3nttQbB8idiHT7YzOKZM\nYVWmkOVSmVm1yhbKO/74/I/fpw+w//7ZLWJ36622unYug+GrO/lkGzeTjbIyG0fXuXP+x6fIY5ih\n4M/ABskAAA49SURBVO2yiy2ylQ0vupiS6tcHrr0WuPnmzF7/4IM2YHf33b05/sknM8zQlvr2BSZP\nzq778Y03LIC0bOlNG37/e9tbaf36ul/70Uc2C+n887059jHH2Hi2FSsyf89339m1JNMVh6mgMcxQ\n8LKtzKjmNyU7ncsvt8rMJ5/U/rpVq2yw7rBh3h27f3+gvDy7KeqTJ2e34jBFS/v2tiHq4sWZv8fL\ngA/YeLJddql77Iyq3QjceKPdGHihWTPg4IOB11/P/D2TJtm1hAgMM+RC9+4WEioqMnv9pEk2AHbP\nPb1rQ8OGFlCuu672dV9uuw04/PDM94HKRL162ZfVv/3W2zZQ+Oy6a+ZjqdavtxlEJ53kbRvuuAP4\n299qPzfHjLHQ5VVVJumUU7I7JxhmKAXDDAVPxKoM336b2euTgxy9LicPGmR3mTXdiU6YADz2mPc7\nbgPZjZtZu9aqOLvt5n07KDz69ct8M9R337Vwm+1U6rr07Wszm37zm/TPV1TY7KX77vOuKpN0wgm2\n5ci6dZm9ftIkBnz6H4YZcmPvvYGvvsrstaNH25RmrxUV2U7Wt9yy9Z5JS5faMe+913bo9tohh9gi\neJmsfDpxIrDTTtnte0PRs/fewPjxmb3W627XVH/7m52bDz+85eMbN9oNwGmnZb91QibatQP22AN4\n5526X7t+vY2Z2WMP79tBkcQwQ270728zlOry/fcWLHJZMTcTO+xgMzhOOsnWkdmwwRblO/BA4Nxz\n7S7VD/Xr2yyUl16q+7Xjx9tdOxW2fv0yCzOVld4sU1CTJk2A116zkH/bbcDPP9tg34EDrYs01xW4\nM5Hp4PiJE219GS5VQAkMM+TGvvtmtpJv8g7Ui+mfNTnmGFt87P77bSzNOecAf/iDXcz9lOkYgfHj\n7a6dCtt22wErV9Y9CPiTT4AuXWzlX79sv73dbPz3v9aVtc8+9mf06Nz2gMrUSSfZubhpU+2v+/xz\nu4YQJTDMkBu9elmpeF6N222ZUaO8WRSsLvvuax8SmzbZTrwXXOD/MY880j4sli6t/XUffQTst5//\n7SG3RKwC+fHHtb9u5Ejr6vFb165WJVm50mbfDRvmb5ABbG+qHj3sd742n3/uX7WWIolhhtwQsamg\nH35Y82umTLE9Z4LcNNTPClB1jRpZoKmtOjN7ts384kDHeDjssNo3bVy/HnjhBeC884JrU7Fv+xGn\nd/bZwNNP1/y8ql039t8/uDZR6DHMkDtHHWULf9XkySdtwGHQF9MgXXgh8PjjNT//3nvAoYdyYbC4\nOOwwm6lUk9des1ltueyuHRXnnWdjyVauTP/8d9/ZmDMvVgOngsEwQ+4ccwzw1lvp13nZuBH497+B\nwYODb1eQjj4amD+/5vVF3nnHwgzFw+67W7djTbPcRozwfn2XsOnQwWb7/ec/6Z9/8027djDgUwqG\nGXKnZ0+gbVvgyy+3fm7sWBtXs8MOgTcrUEVFwCWXAI8+uvVzq1db5WrgwODbRW7UqweceCLw/PNb\nPzd7to3r8mOZgrAZPNjOiXTbO7z2moUZohQMM+TWoEG2MF0qVVvrwouNHaPg4ouB557beiDwyy/b\nwF+vF0ajcBs0yKqS1Q0fbsG3WbPg2xS044+3QcfVx9RNnw5Mm2Zd1EQpGGbIrYsushlLy5dvfmzs\nWOsv92sdjbDp2hU480wLcKkeeaTwuxRoawcfbOdDasVy5kwLODWtzFtoiopseYSbbtqyG/qhh+ya\n4fesKoochhlyq0MHm3qdXNNl2TLbBPLee4OdWeTaTTfZh1VySuqrr9pYmiCm4FK41KsH3HCD7WJd\nWWl/fvUr+7pzZ9etC86gQbaIZXK7kZkzbZbT5Ze7bReFkmg2W86HlIhoIfwcsVVWZuu8HHqoLdR1\n3HG24V3cvPmmrW9z1lnW7TR6NHDQQa5bRS5UVtrg8OLizYHmzTe93w8p7L7/3ipVJ59s+zZdcw0w\nZIjrVlEARASqmvEob4YZCofFi22mRp8+NgAyrjMVvv7aBv0ef7zNbKH4WrvWzol69WwKf1y7VubO\nBZ55BthzT46ViRGGGSIiIoq0bMNMjAYlEBERUSFimCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJI\nY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhj\nmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghIiKiSGOY\nISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhzEmZE5HQRmSwi\nlSKyVy2vmy0i34rINyLyZZBtJCIiomhwVZmZBOBkAB/W8ToFUKKqe6rqPv43K1pKS0tdN8EZ/uzx\nxJ89fuL6cwPx/tmz5STMqOo0Vf0+w5eLr42JsDj/ovNnjyf+7PET158biPfPnq2wj5lRAO+IyHgR\nudR1Y4iIiCh8iv36xiIyDkDHNE/9UVVfzfDbHKCqC0WkHYBxIjJNVT/yrpVEREQUdaKq7g4u8j6A\na1X16wxeezOAVap6V5rn3P0QRERE5DlVzXiYiW+VmSykbayINAZQpKorRaQJgCMBDEv32mx+YCIi\nIiosrqZmnywicwH0BzBWRN5IPN5ZRMYmXtYRwEciMgHAFwBeU9W3XbSXiIiIwstpNxMRERFRvsI+\nm6lWInK0iEwTkR9E5HrX7QmKiHQTkfcTCw9+JyJXuW5T0ESkKLGYYqaDyQuCiLQUkVEiMlVEpohI\nf9dtCoKIXJP4XZ8kIs+KSAPXbfKLiDwhIotFZFLKY61FZJyIfC8ib4tIS5dt9EsNP/vfE7/vE0Xk\nRRFp4bKNfkn3s6c8d62IVIlIaxdt81NNP7eIXJn4//6diNxR1/eJbJgRkSIA9wE4GkBfAGeLyE5u\nWxWYjQCuUdWdYV11V8ToZ0/6DYApsOn7cXIPgNdVdScAuwGY6rg9vhORLgCuBNBPVXcFUATgLLet\n8tWTsOtaqhsAjFPVPgDeTXxdiNL97G8D2FlVdwfwPYA/BN6qYKT72SEi3QAcAeCnwFsUjK1+bhEZ\nAGAggN1UdRcAd9b1TSIbZgDsA2CGqs5W1Y0ARgI40XGbAqGqi1R1QuLvq2AfaJ3dtio4ItIVwLEA\nHkOMFlVM3JEepKpPAICqblLVFY6bFZRiAI1FpBhAYwDzHbfHN4nlJ5ZVe3gggBGJv48AcFKgjQpI\nup9dVcepalXiyy8AdA28YQGo4f87ANwN4LqAmxOYGn7uywH8LfHZDlVdUtf3iXKY6QJgbsrX8xKP\nxYqI9ASwJ+wkj4vhAH4PoKquFxaYbQEsEZEnReRrEXk0MeuvoKnqfAB3AZgDYAGA5ar6jttWBa6D\nqi5O/H0xgA4uG+PQRQBed92IoIjIiQDmqeq3rtsSsN4ADhaRz0WkVET2rusNUQ4zcete2IqINAUw\nCsBvEhWagicixwMoU9VvEKOqTEIxgL0APKCqewFYjcLtbvgfEWkFq0z0hFUgm4rIuU4b5ZDarI3Y\nXf9E5E8ANqjqs67bEoTEjcofAdyc+rCj5gStGEArVe0Pu3F9oa43RDnMzAfQLeXrbrDqTCyISH0A\nowH8W1Vfct2eAO0PYKCIzALwHIBDReRpx20KyjzYXdpXia9HwcJNoTscwCxVLVfVTQBehP0exMli\nEekIACLSCUCZ4/YESkQuhHUtxynEbgcL8BMT17uuAP4rIu2dtioY82DnORLXuyoRaVPbG6IcZsYD\n6C0iPUVkGwBnAnjFcZsCISIC4HEAU1T1H67bEyRV/aOqdlPVbWGDQN9T1fNdtysIqroIwFwR6ZN4\n6HAAkx02KSg/AegvIo0Sv/uHwwZ/x8krAC5I/P0CALG5gRGRo2F35yeq6jrX7QmKqk5S1Q6qum3i\nejcPwF6qGocg+xKAQwEgcb3bRlXLa3tDZMNM4g5tCIC3YBe251W14Gd2JBwA4DwAAxLTk79JnPBx\nFLdy+5UAnhGRibDZTH913B7fqeqXsCrU1wCSYwcecdcif4nIcwA+BbCDiMwVkcEAbgdwhIh8D7vI\n3+6yjX5J87NfBOBeAE1h+/N9IyIPOG2kT1J+9j4p/99TFeS1roaf+wkAvRLTtZ8DUOcNKxfNIyIi\nokiLbGWGiIiICGCYISIioohjmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5ghImdE5CQRqRKR\nHfL4HheKyL1etouIooVhhohcOhvAR4n/5oqLZRHFHMMMETmR2Cj1AACXwLamgIiUJHbJ/Y+ITBWR\nf6e8/tjEY+NF5J8i8mryqZTXtBORUSLyZeJP3PZwIoqlYtcNIKLYOhHAG6r6g4iUi0hy08w9APQF\nsBDAJ4lA8jWAhwAcpKo/icizSF+RuQfAcFX9RES6A3gz8b2IqIAxzBCRK2cDSG6U+nzi69cAfKmq\nCwBARCYA2BbAGgAzVfWnxOufA/DLNN/zcAA72X6UAIBmItJYVdf48yMQURgwzBBR4ESkNYABAHYR\nEQVQBKu0jAWwPuWllbDrVPUqjCA9AbCvqm7wtsVEFGYcM0NELpwG4GlV7amq26pqdwCzAByc5rUK\nYDpsF90eicfORPpuprcBXJX8QkT28LbZRBRGDDNE5MJZAMZUe2x04vGtQoqqrgPwawBvish4AD8D\nWJF8OuU9VwHYW0QmishkpO+KIqICI6qc1UhE4SciTVR1deLv9wP4XlXvcdwsIgoBVmaIKCouFZFv\nEhWX5gAedt0gIgoHVmaIiIgo0liZISIiokhjmCEiIqJIY5ghIiKiSGOYISIiokhjmCEiIqJIY5gh\nIiKiSPt/CAFWfGPEh+sAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7d7d320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_fourier(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even though it's better, it doesn't look very much like the square wave, does it? We can set $n$ as an interactive parameter, and we can observe how modifying it affects our function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DTHicM0Pu8TCTL5wAHB7DjC8M+OHxMBO5x8NMfqjaHwt3WAwzvjDgh8fDTOQe\nC7cfUZBhmAmLfcIXHmYKj4eZyD0eZvKDRdsHfmmeL+wX4fEwE7nHw0x+cG6AD/zSPF8YZsLjYSZy\nj0PqfnBugA/sE74wzITHw0zkHg8z+cGi7QPDjC/sF+HxMBO5x8NMfrBo+8Aw4wv7RXgMM+QeR2b8\n4JwZHxhmfGGYCU8kbfPIuGYpHizcfnDOjA/sE74wzISXxj7BNUvx4GEmP1i0fWCY8YX9Irw0fsKP\na5biwcNMfrBo+8DvmfGFh1/D48gMuceRGT8YZnzg98z4wsOv4aUx4HPNUjw4pO4H34H6wD7hC0N+\neDzMRO7xMJMffAfqA8OMLwwz4fEwE7nHw0x+sGj7wDDjC/tFeAwz5B4Ltx8s2j6wT/jCfhEe58yQ\nezzM5AfnzPjAMOMLw0x4nDND7vEwkx+cM+MDw4wvDDPh8TATucfC7QeLtg/sE76wX4SXq2FGRE4Q\nkWkiMlNErqng+kIRWS0iExN//wzRTqoEHmbyg0XbB35pni/sF+GlsU9UScujVoKIFAB4CEA3AAsB\nfCoib6nq1M1u+r6q9sh4A2n78AvC/GDR9oF9whf2i/BydM5MZwCzVHWuqn4PYBCAUyu4HWcyZgMO\nqfvBCcA+sE/4wn4RXo4eZmoOYH7K+QWJy1IpgMNFZJKIDBORjhlrHW0fFm4/OAHYB/YJX9gvwktj\nnwh2mAkWVLZlAoCWqrpBRE4EMARA+/Q2i3YI5wf4weF0HxhmfGG/CC8X58zA5sm0TDnfEjY68wNV\nXZuyPFxEHhGRXVV1xeYP1rdv3x+WCwsLUVhYGHd7aWs4P8APFm0fGGZ8Yb8Ibyv7iaKiIhQVFe3w\nQ4cMM+MBtBORNgAWATgLwNmpNxCRJgCWqqqKSGcAUlGQAX4cZigAFm4/WLR9YJ/whf0ivK30ic0H\nIfr167ddDx0szKhqqYhcCmAEgAIAT6nqVBG5KHF9fwA9AVwsIqUANgD4Taj20jawcPvBiY4+sE/4\nwjATXo7OmYGqDgcwfLPL+qcsPwzg4Uy3i3YA58z4wYmOPrBP+MIwEx5/m4nc45wZP1i0fWCf8IX9\nIrwc/Z4ZyiUcUveDRdsH9glf2C/Cy9HvmaFcwsLtB+fM+MA+4QvDTHg8zETucX6AH5wz4wPDjC8M\n+eHxMBO5x/kBfvAdqA8MM76wX4THw0zkHgu3HyzaPrBP+MIRy/AYZsg9Fm4/GGZ8YJ/whf0iPM6Z\nIfc4Z8bELXxHAAAgAElEQVQPzg3wgX3CF4aZ8DhnhtzjnBk/OJzuA0dmfGGYCY+Hmcg9Fm4/WLR9\nYMD3I1oPHLEMi2GG3GOY8YNhxgf2CT/YJ3zgnBlyj/MD/GDh9oFhxg/2CR84Z4bc45C6H5wA7APD\njB8MMz7wMBO5x8LtBycA+8A+4QfDjA8MM+QeC7cfLNw+sE/4wdFKH0R4mImc45wZPxhmfGCY8YN9\nwocoUKYh0HDtUjw4Z8YPvgv1gQHfDx569SNNIZ9rl+LBd6F+sHD7wIDvB0dm/GCYIdf4LtQPFm4f\nGPD9YJ/wI037Cq5digffhfrBwu0Dw4wf7BN+pGlfwbVL8WDh9oOF2wf2CT/YJ/zgYSZyjYXbD04A\n9oF9wg+GGT8YZsg1zpnxgxOAfWCY8YNhxg/OmSHXOGfGDxZuHxhm/GCf8INzZsg1Fm4/WLh94Gil\nH+wTfvAwE7nGMOMH58z4wNFKP9gn/OBhJnKN70L94LtQHxjw/eA8Mj94mIlc47tQP1i4fWCY8YMB\n349Qh5lEpKaI3CAiTyTOtxORk2NvCWU3Fm4/WLh9YJ/wg33Cj4BzZp4BsAnA4YnziwD8K/aWUHZj\n4faDhdsH9gk/2Cf8CDhnZk9V/Tcs0EBV18feCsp+nDPjByc7+sAw4wfDjB8B58xsFJHq0RkR2RPA\nxthbQtmNc2b84JwZHxhm/GCY8SNN/aJKJW7TF8B/ALQQkZcAHAHg/NhbQtmNhdsPFm4fOFrpB/uE\nH6HCjKqOFJEJAA5NXHS5qn4Xe0souzHM+MHC7QP7hB/sE36kKeRvMcyISCcAqccNFgEQAK1EpJWq\nToi9NZS9+C7UDxZuH3jo1Q/2CT/S1C+2NjJzNyzMVAfQCcAXicv3AzAewGGxt4ayFwu3H5wA7ANH\nZvxgmPEj0x/NVtVCVe0CG5E5SFU7qWonAAcmLiNKYuH2gxOAfWCf8IMB34+A3zOzt6pOjs6o6hQA\nHWJvCWU3Fm4/+C7UB/YJPxjw/cj0nJkUX4jIkwBegM2ZOQfApNhbQtmNc2b8YJjxgWHGD/YJPwLM\nmYn0AnAxgCsS5z8A8GjsLaHsxjkzfnBI3QeGGT8YZvwI+NHsYgD3JP6IKsbC7QeH1H1gn/CDYcaP\nUGFGRL6u4GJV1T1ibw1lLxZuP1i4feChVz/YJ/wIOGfmlynLuwDoCaBB7C2h7MbC7QcLtw889OoH\n+4QfoX6bSVW/S/lboKr3Aegee0sou7Fw+8HC7QNHK/1gn/Aj4GGm1G8C3gnAwQAKYm8JZTcWbj84\nAdgH9gk/GGb8CHiYKfomYAAoBTAXwP/F3hLKbizcfqgCBXy/EVwUKFUZLkNjmPEj4K9mX6Cqc1Iv\nEJHdY28JZTfOmfGDhduPqHAzXIbFPuFHqDkzAF6r5GXbTUROEJFpIjJTRK7Zwm0eSFw/SUQOjOP/\nUhpwzowfLNx+cMTSBx569SPTIzMi0gFARwD1ROTXsG//VQB1YJ9q+llEpADAQwC6AVgI4FMReUtV\np6bc5iQAbVW1nYgcAvuyvkN/7v+mNGDR9oNhxg/2Cx/43Ut+BJgzsxeAUwDUTZxG1gL4Uwz/uzOA\nWao6FwBEZBCAUwFMTblNDwADAEBVPxaReiLSRFWXxPD/KU4s2n7wXagfPPzqAwO+H5n+OQNVHQJg\niIgcpqofxf6fgeYA5qecXwDgkErcpgUAhhlvWLT94LtQPxjyfWCY8SPAYaZrVPXfAM4RkXM2u1pV\n9fKf+b8rG802f4tZ4f369u37w3JhYSEKCwt3qFG0gzhnxg8Wbj/YL3xgn/CjUSOgyk+jR1FREYqK\ninb4Ybd2mOmrxOlnFVwXR+9cCKBlyvmWsJGXrd2mReKyn0gNMxQA34H6wcLtB/uFD+wTfgweXOHF\nmw9C9OvXb7sedmuHmYYmTp/drkesvPEA2olIGwCLAJwF4OzNbvMWgEsBDBKRQwGs4nwZp3iYyQ/O\nmfGDYcYHhpmct7XDTEO3cj9V1R4/5x+raqmIXApgBOwbhZ9S1akiclHi+v6qOkxEThKRWQDWA+j1\nc/4npRGLth+cM+MH+4UPDDM5b2uHme7eynWxHARW1eEAhm92Wf/Nzl8ax/+iNOPcAD9YuP1gmPGB\nfSLnbe0wU1G0LCLVAOwNoBzAdFXdlP6mUVZh0faDhdsP9gsf2CdyXmV+aLI7gMcARD9psIeIXKSq\nw9LaMsounDPjBwu3HwwzPrBP5LzK/DbTPQC6qOosABCRPQEMS/wRGR5m8oMTgP1gyPeBfSLnVSaq\nromCTMIcAGvS1B7KVnwH6gcnAPvBkO8D+0TOq8zIzGciMgzAK4nzZwIYn/i9Jqjq6+lqHGURhhk/\nOKTuB/uFD+wTOa8yYWYXAEsBHJ04vyxxWfR7TQwzxOF0T1i4/WCY8YF9IudtM8yo6vkZaAdlOw6n\n+8H5AX4wzPjAMJPzKvNppj0AXAagTcrtf/aX5lGOYdH2g4XbD/YLH9gncl5lDjMNAfAkgKGw75kB\nYvrSPMohLNp+cLKjH+wXPjDM5LzKhJliVX0g7S2h7MY5M36wcPvBMOMD+0TOq0yYeUBE+gAYCWBj\ndKGqTkhbqyj7cM6MHyzcfjDk+8A+kfMqE2b2BfA7AMcgeZgJALqkpUWUnfgO1A9OAPaDId8Hhpmc\nV5kw838A9uDvMdFWMcz4wTkzfrBf+MAwk/Mqs3YnA6if7oZQluNwuh8s3H4wzPjA0cqcV5mRmfoA\nponIp0jOmeFHs+nHOJzuB8OMHwwzPnC0MudVJsz0SZwqAAFwFIDfpK1FlJ1YtP1gmPGD/cIH9omc\nt821q6pFsB+WPBnAANhE4EfT2yzKOizafnBI3Q/2Cx8YZnLeFkdmRGQvAGfDRmGWA3gZgKhqYWaa\nRlmFc2b84JC6HwwzPjDM5LytHWaaCmAMgFNUdSYAiMhVGWkVZR/OmfGDhdsPhnwf2Cdy3tbW7q8B\nfAtgtIg8ISLdYHNmiH6K70D9YOH2g/3CB/aJnLfFtauqQ1T1LAB7AygCcCWARiLyqIgcl6H2UbaI\n5mhwdCY8zpnxgyOWPjDM5LzKTABep6ovqurJAFoCmAjg2rS3jLIPh9R94JwZPzgy4wPDTM7brrWr\nqitU9XFVPSZdDaIsxnehPrBw+8Ew4wP7RM7j2qX4sHD7wMLtB/uEDzz0mvNY8Sg+LNw+MMz4wT7h\nA/tEzuPapfhwzowPfBfqB8OMD5xHlvO4dik+nDPjAwu3HwwzPnBkJudx7VJ8WLh9YOH2g6OVPrBP\n5DyuXYoPw4wPLNx+cLTSB/aJnMe1S/Hhu1AfWLj9YMD3gX0i53HtUnz4LtQHTgD2g2HGB4aZnMe1\nS/Fh4faBE4D9YJ/wgWEm53HtUnxYuH1g4faDfcIH9omcx7VL8eGcGR9YuP1gmPGBfSLnce1SfDhn\nxgfOmfGDYcYH9omcxzBD8WHh9oFzZvzgaKUPHJnJeVy7FB8Wbh9YuP1gwPeBAT/nce1SfHiYyQeG\nGT/YJ3xgn8h5XLsUH74L9YGF2w/2CR/YJ3Ie1y7Fh4XbB0529IN9wgeGmZzHtUvx4ZwZHzg/wA+G\nGR8YZnIe1y7Fh/MDfODIjB8MMz4wzOQ8rl2KDwu3D+XlQEFB6FYQwD7hBcNMzuPapfiwcPvAwu0H\n+4QP7BM5j2uX4sM5Mz6UlbFwe8E+4QPDTM7j2qX4cM6MDzzM5Af7hA+cR5bzqoT4pyKyK4CXAbQG\nMBfA/6nqqgpuNxfAGgBlAL5X1c4ZbCZtLw6p+8B3oX6wT/jAT/jlvFBr91oAo1S1PYD/Js5XRAEU\nquqBDDJZgIXbB4YZP9gnfGCfyHmh1m4PAAMSywMAnLaV23JsMFtwfoAPLNx+MMz4wD6R80Kt3Saq\nuiSxvARAky3cTgG8KyLjReRPmWka7TDOD/CBhdsPhhkf2CdyXtrmzIjIKABNK7jq+tQzqqoisqU9\n4BGqulhEGgEYJSLTVHVM3G2lmLBw+8DC7Qf7hA/sEzkvbWFGVY/d0nUiskREmqrqtyKyG4ClW3iM\nxYnTZSLyBoDOACoMM3379v1hubCwEIWFhTveeNoxLNw+sHD7wT7hA/uEe0VFRSgqKtrh+wf5NBOA\ntwCcB+DfidMhm99ARGoAKFDVtSJSE8BxAPpt6QFTwwwFwjkzPrBw+8E+4QP7hHubD0L067fF3X2F\nQq3d2wEcKyIzAByTOA8RaSYi7yRu0xTAGBH5HMDHAN5W1ZFBWkuVwzkzPvBL8/xgn/CBYSbnBRmZ\nUdUVALpVcPkiAN0Ty3MAHJDhptHPwSF1H/ileX6wT/jAMJPzuHYpPizcPrBw+8E+4QP7RM7j2qX4\ncH6ADyzcfjDM+MCfM8h5rHgUH84PCC96/Vm4fWCY8YE/Z5DzuHYpPizc4XFUxhf2CR/YL3Ie1y7F\nh4U7PBZtX9gnfGC/yHlcuxQfzpkJj0XbF/YJH9gvch7XLsWHc2bCY9H2hSMzPrBf5DyuXYoPC3d4\n/MI8XxjwfWCYyXlcuxQfhpnw+IV5vrBP+MAwk/O4dik+nB8QHou2LwwzPrBf5DyuXYoPh9TDY9H2\nhWHGB/aLnMe1S/Fh4Q6PRdsX9gkf2C9yHtcuxYeFOzwWbV/YJ3zgzxnkPFY9ig/nzITHMOMLw4wP\n/DmDnMe1S/HhnJnwGGZ8YcD3gf0i53HtUnz4LjQ8Fm1fGPB9YL/IeVy7FB++Cw2PX5rnCwO+Dwwz\nOY9rl+LDwh0evzTPF/YJHxhmch7XLsWHQ+rhsWj7wjDjA/tFzuPapfiwcIfHou0L+4QP7Bc5j2uX\n4sM5M+GxaPvCMOMD+0XO49ql+PAwU3gs2r4wzPjAfpHzuHYpPizc4bFo+8LRSh/YL3Ie1y7Fh2Em\nPBZtX9gnfODPGeQ8Vj2KD9+Fhscw4wsPvfrAnzPIeVy7FB++Cw2PX5rnC/uED+wXOY9rl+JTUGBF\ng8Lhl+b5wjDjQ1kZ+0WOY5ih+DDMhMfDTL4wzPjAMJPzWPUoPgwz4THM+MIw4wPDTM5j1aP4VKnC\nMBMaw4wvDDPhqfLwax5g1aP4cGQmPIYZXxhmwov6BD+andNY9Sg+BQVAaWnoVuQ3hhlf+HUF4ZWW\nclQmD7DqUXw4MhMew4wv/J6Z8DhfJi+w6lF8GGbCY5jxhYeZwmOYyQusehQfhpnw+OVgvjDMhMcw\nkxdY9Sg+/DRTePzUhi8MM+GVlVltopzGMEPx4chMeDzM5AvDTHgcmckLrHoUH36aKTyGGV8YZsLj\np5nyAqsexYcjM+ExzPjCMBMeR2byAqsexYdhJjyGGV/4PTPhMczkBVY9ig8nAIdTXAwsXMgw4030\nPTPl5cCcOaFbk584ATgvsOpRfDgyE85VVwEtWjDMeBMdZnr9dWDPPUO3Jj9xZCYvsOpRfKIJwEOH\nAmecEbo1+WXGDDtlmPElCjPffx+6Jfln5EigRw9g0yaGmTzAqkfxiUZm+ve3d6KUOWvX2im/NM+X\nKMxE62TNmrDtyScff2xvrL79lmEmD7DqUXyiMFO1qp3nxMfMicIMvzTPlyjMrF5t5xcsCNuefFJc\nbKfr1rFP5AGGGYpPNAF40yY7v25d2Pbkk9QwIxK2LZS0007WJxhmMi8aBVu3jhOA8wDDDMUnGpmJ\n3hGtWgUcfTTwwANh25WrBg4E6tWz5WgUrLSUhduTqE9EYWbVKuDNN5OjlxQvVQvzixYlw8zatRyZ\nyQNBwoyInCkiX4pImYgctJXbnSAi00Rkpohck8k20g6ICveGDXZ+1Srggw+AO++08998A5SUhGtf\nrvnqK9tJpo6GrV0L7Lxz2HZR0s472+TfKMysXQu88w4nBMdt5kwLMpMn2/l58xhm8kyokZnJAE4H\n8MGWbiAiBQAeAnACgI4AzhaRDplpHu2Q6NNMxcVAzZo/HVpv3Rq45RZbXr/eig9Vzvz5FlqmTwe6\ndAGWLk0eWpo+3Zbr1gW++45hxpPUMFO3rq2nqD+UlAC33grceKOd/+YbfrXB9li3zmqIKtC+PTB8\nODBunF23cKGFmZ124pyZPBEkzKjqNFWdsY2bdQYwS1Xnqur3AAYBODX9raMdlnqYqWlTYMUKu1zE\nRmkA2ykvWADUqgW8/bad79PHQtDy5TaSA9jj5Pqcm9QwF71Tnz3bAgkA/PnPwLvvWlFu1Qp49llg\nzBigqMhOv/7abjd3rp02asQw400UZtasAZo3t9MlS+y6efOA668Hbr7ZzrdubcuqQLdutl5VgU8+\nSX7xXvTbZ9FOPJdFIyurVwPvvWfLY8YATzxhy7VrA3/9q/UZwILM4sW2vGCB3a9JE4aZPOF5zkxz\nAPNTzi9IXEZeRROAi4uBZs2syDRoANSoAYwda++SJk8GPvzQbv/xx/YR7ptusndVvXvbHJtVq4AT\nT7R3W6pA9+7A3XcDGzcCnTvb/b77DjjrLDudOhW44QYr9EVFwEsv2eO/9ZYVP1XglVdsBOP774FB\ng+x/rF4NPPOM7SQWLgRee83uN2mS7UAA+2jn7Nn2GE8/DSxbZqNKjz1mh9NmzQKeesoe44MPgOef\nt/s9/7w9XlmZPa///c/+Z+fO1o5x4+z1WLHCdmBVq1r7u3Sx4LJmjX3EvW9fe04A8MUXwLRptjx7\ntu0M99zTdnq1a9vrvGYNw4wnUZgpLrYd69q1Ftr33NO2HRHbDqJvBx41yraV//4XePFF25YPOcT6\nzx132OOtXg1cfDHwq1/ZfS65BHjwQfs/11wDfPqp3eaKK+x/LVpk21F5ufW/F16w+40cmdzOX3nF\nwnFZmV2/YoW1dfBge9xvvgGGDLHbTp5sIRuwdn71lfWPJ5+0MBGNOG3YYPe79Vb73yNHWr8BgHvu\nAQYMsP932mnAq69aADn4YAsuI0faSNaMGcB11wFdu1p7unUDLrww+eZoxAhg/Hh7DSdOtI9ht25t\nj7Vmjb2p4gTgvJC2NSwiowA0reCq3qo6tBIPkeNvO3JQ6sjMbrvZjrdpUytun35qhWjsWODLL4H9\n97ciVL++FZ9x46yAtWljAeeDD6wAvfYaMGyYBZjate1xeve2+7/yCtCyJTBhAjB6NNCxI3DttVZA\nGzYETj3V2vTGGxZ8jj7avszv8suBXr2sAD71lBXdp56yYjh0KHDOOdbm0aPtS7eid8x/+IM95l57\n2Y5l9Wrg/fctiBUUAFdeaZc1bgz8/vfW/gEDgNtus+d27rnW/ttvT45aPfmkhaSqVS18rVxp7Xrl\nFXuOU6ZYiOnc2XY8DRoAxx1nO79Fi4AjjrAwU6cOsMsuVvCbM/O7EYWZkhLrE1GY6drVtulWrYBq\n1Sxgd+5sfaOoyNb9qFG2fTdqZH1g1CjbToYMsVG6jRtt237kEXuMevVsu/zwQ3v8Bx6wkDFvnm1b\nLVvatjdrlv3fk06yMDV0qPWPffYBLrvMRgQvucTa//DD9phDhljffest4OyzLdAPH25vOlq3tjD1\npz8Bxx9vgeRf/7Jtefp0e/x27YC//c36ZtOmwNVX2+PXqGEToqdMAc47D/jsMwvxNWta215+2Z57\nmzY24b1qVasDb74JdOpkr9eUKVZb5s+3PnfAARZq1qwB2rblnJl8oarB/gCMBnDQFq47FMB/Us5f\nB+CaLdxW+/Tp88Pf6NGjlQJ47z3VwkLVGjVUL79ctWtX1aOOUj3iCNXu3VWvvlq1fn3VX/1K9a67\nVBs0UG3ZUvWBB1T32MPO33mnatOmqscco/rHP9pg+vXXqx55pC0//7zqTjvZ8jvv2Gm9eqqvvWbL\nhx2mevPNtnzZZapnnGHLTzyh2q6dLRcVqVatqiqiOmKEXdahg90GUL3wQtUrr7Tlf/0r+b/feEO1\nVi1b/s9/7LRZM9W337blY45R/ec/bfmKK1RPOSXZ5hYtbHnQoOgAgerAgapVqqjuvrtqv36qTZrY\nfU49VXXPPVV797b7/fKXqvfco1qnjr1O991nr3OVKqrXXqt65pmqv/iFva5RG8iHNWtUa9ZU7dxZ\n9dJLVXv2tPX297/buurSRbVHD9WDDrLLmjZVPfhg1UcesW2taVPVp55SbdTI+s6DD9r9u3RR/ctf\nbDu66ip7LED1pZdsO0rdRkVUX3/dljt3tj4GWP867zxbfuQR1X33teWhQ60PV62q+uGHdlnbtqrP\nPWfLPXuq/u1vtnzPParHHWfLL76ouuuutjxqlJ3WqGGXA1YH7rnHli+6yLZzQPX++60PAKrDhqkW\nFKjuvLM976ZNVRs3Vr3tNusTXbqonnaavV5XXKHasaPq/vur3nqr/e9DD7V+c8IJ9hgXXKB68sl2\nnlwbPXr0j/bjFk+2I09sz43j/kuEmU5buK4KgNkA2gCoCuBzAB22cNtYX1TaQe+/bzt+ESs+bdqo\nnn66Fb+GDW0nfPDBttlNm5bcqa9fb6cnnKA6d64tP/OM6tixFlymTrVifMwxqhs2WOj4+9/tfz76\nqOqrr6qWldnjf/mlanGx6k03qa5erfrdd6r33mvXT59uAUjVAs348bY8ZIjqnDl2m4ED7T5r19rj\nbtyo+s03qi+/bLf9/HMr9qpWeD/7TLW83Ir03Ln2XK6+WnXlSnuOV1xhjzFokAWP8nLVc8+1y0tK\nrNj36aM6eXIy+Dz5pC2/+67qr39ty7Nnq9aubctTpqhWq6a6226qd99tr+nhh6see6wFn379MrnW\naWuKi22nut9+qn372rpq3Ni2yWrVVP/wB9V//CO5zUc7+Fmz7HpA9fvv7fS3v1VdtcqW33xT9euv\nVatXV50/X3XcONXf/1510ybrKw8+aP//mWcssJeXq/bvb9tZcbEFhRUrbFt//HHV0lLVefNUX3nF\n7jdxouonn9jy0KHWP8rLVZ991u63fr09RkmJ6sKF1lfKy1U/+kj1rbfsfkOG2DZcVmaBZcoUC3cX\nX2z3mTrVwsmGDaovvKDarZs9Ro8ett1v2mTPtVcv62eAhZY77kj2lbPPtuX//tder4YNLdB17Gh9\n669/VT36aHszRVklK8IM7JNM8wEUA/gWwPDE5c0AvJNyuxMBTAcwC8B1W3m8uF9H2hEffqjaqZMV\n7/799Yd3f9dfnyzAUfEpK1O9/Xb7U7VitnChLX/5pRXXfFBaagVc1QKUqoWw/v1t+Y03Et1Uk+9e\nS0rsdP/9befSsKEFwVNOUd17bwuS5ENpqa2r9u1t9KNhQxsFfPll/WHkLwqvEyaoPvywjcCp2s5/\n2DBbLi62UKOaPM0HM2ZY2FG1bX3pUntTctppVi+iEdQVK5Kjn9OmWQ1q3txqT6dOFpAoq2xvmAky\nK0pV3wDwRgWXLwLQPeX8cADDM9g0+jmqVLHJdtWrJ7/MrUEDYO+9bfnQQ20uwGGH2byQa1K+Oui3\nv00ud+yYuTaHlnosv1YtO61TxyY5AjY5Mvq47ocf2mTnatXsfO3ayU8w1a9vE4j5PTO+FBTYtr5+\nvc2Z+e47276jbbxZM5uX1bQpcOCB9vfnP9t1++xjf4DNh4rk02TWdu2Sy+edZ6eNGtlcIcDm9uy1\nl23/devaxN/GjW2e0q67Wi3iBOC8wDVM8SkosJ1pjRo/DjNRQW7c2E4vuyxM+7JV9COFzZrZH2Af\nZ+/ZM3mbVq1sQjDDjD8772zrpX17O9+qFbDvvjZhvGtX2+l27568PX8otPLatrU/wCYmv/yyhRog\n+Qk/TgDOC+w1FJ+Cgp+OzDRqZO82lR9Oi1XfvrZDbN3azkefimGY8ScKM7vtZuejvnHvvfYJI4rH\n9dfbJ/+iMFhcnByZYZjJeQwzFJ/UMBO9OzrppLBtynW1a9tpzZp2KEKVYcabnXe29RIdKuIPgabf\nHnvYIT2GmbzBMEPxKSiwL8eqXt2G1L/6KnloidKnYUPgoIOSO0uGGV+i+RrVqtl3oJx4Ytj25IMJ\nE4CPPrJaVF7OMJMHOGeG4hMV7erV7d1nB/6UVkYsW2anDDM+RYdYo2+ppfSrW9f+qle385wAnPM4\nMkPxid79RAWEMiv6lBPDjC/88chwolrEkZmcxzBD8YkKRo0aYduRrzgy4xPDTDhRLWKYyXkMMxQf\njsyExTDjE8NMOByZyRsMMxQfhpmwGGZ8Ki8P3YL8xTCTNxhmKD7RzpRhJgyGGZ84MhNOVIui+WSU\nsxhmKD7169sp3wWFseuudsow48umTfw0TShRmGnaNGw7KO0YZig+0TdvrlkTth35KirYDDP+NGwY\nugX5KZoAzDCT8xhmKH6rV4duQX6Kvi6fYcafBg1CtyA/cWQmbzDMUPwYZsJo1MhO+UOF/jDMhBEd\n8ubrn/NY9ShehxwCdOkSuhX5KZqXsWpV2HbQT3FnGlaLFqFbQGnGWWkUr3HjQrcgvw0YABx2WOhW\nUKphw+wXzimM6OckKKeJ5sCKFhHNhedBREREgIhAVSv9E/M8zERERERZjWGGiIiIshrDDBEREWU1\nhhkiIiLKagwzRERElNUYZoiIiCirMcwQERFRVmOYISIioqzGMENERERZjWGGiIiIshrDDBEREWU1\nhhkiIiLKagwzRERElNUYZoiIiCirMcwQERFRVmOYISIioqzGMENERERZjWGGiIiIshrDDBEREWU1\nhhkiIiLKagwzRERElNUYZoiIiCirMcwQERFRVmOYISIioqzGMENERERZjWGGiIiIshrDDBEREWU1\nhhkiIiLKagwzRERElNUYZoiIiCirBQkzInKmiHwpImUictBWbjdXRL4QkYki8kkm20hERETZIdTI\nzGQApwP4YBu3UwCFqnqgqnZOf7OyS1FRUegmBMPnnp/43PNPvj5vIL+f+/YKEmZUdZqqzqjkzSWt\njcli+byh87nnJz73/JOvzxvI7+e+vbzPmVEA74rIeBH5U+jGEBERkT9V0vXAIjIKQNMKruqtqkMr\n+TBHqOpiEWkEYJSITFPVMfG1koiIiLKdqGq4fy4yGsDVqjqhErftA2Cdqt5dwXXhngQRERHFTlUr\nPbp+jMQAAAapSURBVM0kbSMz26HCxopIDQAFqrpWRGoCOA5Av4puuz1PmIiIiHJLqI9mny4i8wEc\nCuAdERmeuLyZiLyTuFlTAGNE5HMAHwN4W1VHhmgvERER+RX0MBMRERHRz+X900xbJSIniMg0EZkp\nIteEbk+miEhLERmd+OLBKSJyeeg2ZZqIFCS+TLGyk8lzgojUE5HXRGSqiHwlIoeGblMmiMhViW19\nsoi8JCLVQrcpXUTkaRFZIiKTUy7bVURGicgMERkpIvVCtjFdtvDc70xs75NE5HURqRuyjelS0XNP\nue5qESkXkV1DtC2dtvS8ReSyxHqfIiL/3tbjZG2YEZECAA8BOAFARwBni0iHsK3KmO8BXKWq+8AO\n1V2SR889cgWAr2Af388n9wMYpqodAOwHYGrg9qSdiDQHcBmATqr6CwAFAH4TtlVp9QysrqW6FsAo\nVW0P4L+J87moouc+EsA+qro/gBkArst4qzKjoucOEWkJ4FgA8zLeosz4yfMWkS4AegDYT1X3BXDX\nth4ka8MMgM4AZqnqXFX9HsAgAKcGblNGqOq3qvp5YnkdbIfWLGyrMkdEWgA4CcCTyKMvVUy8I/2V\nqj4NAKpaqqqrAzcrU6oAqCEiVQDUALAwcHvSJvH1Eys3u7gHgAGJ5QEATstoozKkoueuqqNUtTxx\n9mMALTLesAzYwnoHgHsA/CPDzcmYLTzviwHclti3Q1WXbetxsjnMNAcwP+X8gsRleUVE2gA4ENbJ\n88W9AP4OoHxbN8wxuwNYJiLPiMgEEXki8am/nKaqCwHcDeAbAIsArFLVd8O2KuOaqOqSxPISAE1C\nNiagCwAMC92ITBGRUwEsUNUvQrclw9oBOEpExolIkYgcvK07ZHOYybfDCz8hIrUAvAbgisQITc4T\nkZMBLFXVicijUZmEKgAOAvCIqh4EYD1y93DDD0SkPmxkog1sBLKWiPw2aKMCUvvURt7VPxG5HsAm\nVX0pdFsyIfFGpTeAPqkXB2pOplUBUF9VD4W9cX1lW3fI5jCzEEDLlPMtYaMzeUFEdgYwGMALqjok\ndHsy6HAAPUTkawADARwjIs8FblOmLIC9S/s0cf41WLjJdd0AfK2qy1W1FMDrsO0gnywRkaYAICK7\nAVgauD0ZJSLnww4t51OI3RMW4Ccl6l0LAJ+JSOOgrcqMBbB+jkS9KxeRBlu7QzaHmfEA2olIGxGp\nCuAsAG8FblNGiIgAeArAV6p6X+j2ZJKq9lbVlqq6O2wS6Huq+vvQ7coEVf0WwHwRaZ+4qBuALwM2\nKVPmAThURKontv1usMnf+eQtAOclls8DkDdvYETkBNi781NVtSR0ezJFVSerahNV3T1R7xYAOEhV\n8yHIDgFwDAAk6l1VVV2+tTtkbZhJvEO7FMAIWGF7WVVz/pMdCUcAOBdAl8THkycmOnw+yrfh9ssA\nvCgik2CfZro1cHvSTlU/gY1CTQAQzR14PFyL0ktEBgIYC2AvEZkvIr0A3A7gWBGZASvyt4dsY7pU\n8NwvAPAggFqw3+ebKCKPBG1kmqQ89/Yp6z1VTta6LTzvpwHskfi49kAA23zDyi/NIyIioqyWtSMz\nRERERADDDBEREWU5hhkiIiLKagwzRERElNUYZoiIiCirMcwQERFRVmOYIaJgROQ0ESkXkb1+xmOc\nLyIPxtkuIsouDDNEFNLZAMYkTncUvyyLKM8xzBBREIkfSj0CwB9hP00BESlM/EruqyIyVUReSLn9\nSYnLxovIAyIyNLoq5TaNROQ1Efkk8Zdvv+FElJeqhG4AEeWtUwEMV9WZIrJcRKIfzTwAQEcAiwH8\nLxFIJgB4DMCvVHWeiLyEikdk7gdwr6r+T0RaAfhP4rGIKIcxzBBRKGcDiH4o9eXE+bcBfKKqiwBA\nRD4HsDuADQDmqOq8xO0HAriwgsfsBqCD/R4lAKC2iNRQ1Q3peQpE5AHDDBFlnIjsCqALgH1FRAEU\nwEZa3gGwMeWmZbA6tfkojKBiAuAQVd0Ub4uJyDPOmSGiEHoCeE5V26jq7qraCsDXAI6q4LYKYDrs\nV3RbJy47CxUfZhoJ4PLojIgcEG+zicgjhhkiCuE3AN7Y7LLBict/ElJUtQTAXwD8R0TGA1gDYHV0\ndcp9LgdwsIhMEpEvUfGhKCLKMaLKTzUSkX8iUlNV1yeWHwYwQ1XvD9wsInKAIzNElC3+JCITEyMu\ndQD0D90gIvKBIzNERESU1TgyQ0RERFmNYYaIiIiyGsMMERERZTWGGSIiIspqDDNERESU1RhmiIiI\nKKv9P9wCiN/X1UIpAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7d93ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "myPlot = interactive(plot_fourier, Z=(1,100,1))\n",
    "display(myPlot)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can observe the noise around the ends, which is less noticeable as $n$ increases. We expect our function to be equal to the square wave as $n \\rightarrow \\infty$. I chose $100$ as the maximum number of terms because the noise reduction is negligible for higher values of $n$, and it takes too much computing power."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "VPython",
   "language": "python",
   "name": "vpython"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}