{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "electric-edgar",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This tutorial discusses how to define functions within Python code.\n",
    "# Once defined, the functions can be evaluated, differentiated, integrated,\n",
    "# plotted, ...  \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "numeric-narrative",
   "metadata": {},
   "outputs": [],
   "source": [
    "# The notation used to define a function is the following:\n",
    "def f(x):\n",
    "    return np.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "elegant-search",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n",
      "0.7071067811865476\n",
      "1.0\n"
     ]
    }
   ],
   "source": [
    "# You can all your function anything, it doesn't have to be f.  Also, the \n",
    "# variables are defined within the function and do not need to be declared\n",
    "# outside of the function.\n",
    "    \n",
    "# Once defined, it's easy to evaluate the function at any value of x.\n",
    "print(f(0))\n",
    "print(f(np.pi/4))\n",
    "print(f(np.pi/2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "virgin-proportion",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.09983342,  0.19866933,  0.29552021,  0.38941834,\n",
       "        0.47942554,  0.56464247,  0.64421769,  0.71735609,  0.78332691,\n",
       "        0.84147098,  0.89120736,  0.93203909,  0.96355819,  0.98544973,\n",
       "        0.99749499,  0.9995736 ,  0.99166481,  0.97384763,  0.94630009,\n",
       "        0.90929743,  0.86320937,  0.8084964 ,  0.74570521,  0.67546318,\n",
       "        0.59847214,  0.51550137,  0.42737988,  0.33498815,  0.23924933,\n",
       "        0.14112001,  0.04158066, -0.05837414, -0.15774569, -0.2555411 ,\n",
       "       -0.35078323, -0.44252044, -0.52983614, -0.61185789, -0.68776616,\n",
       "       -0.7568025 , -0.81827711, -0.87157577, -0.91616594, -0.95160207,\n",
       "       -0.97753012, -0.993691  , -0.99992326, -0.99616461, -0.98245261,\n",
       "       -0.95892427, -0.92581468, -0.88345466, -0.83226744, -0.77276449,\n",
       "       -0.70554033, -0.63126664, -0.55068554, -0.46460218, -0.37387666,\n",
       "       -0.2794155 , -0.1821625 , -0.0830894 ])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can also input a vector of x-values and evaluate the function at each\n",
    "# value of x contained in the array.\n",
    "xx = np.arange(0,2*np.pi,0.1)\n",
    "y = f(xx)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "disturbed-support",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# These data can then easily be plotted.\n",
    "plt.plot(xx, f(xx), 'bo')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x) = sin x');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "facial-narrow",
   "metadata": {},
   "outputs": [],
   "source": [
    "# If you wanted to do symbolic math using a function, you could use 'sym.sin()'\n",
    "# instead of 'np.sin()'.  This requires the module 'sympy'.\n",
    "import sympy as sym\n",
    "def f1(x):\n",
    "    return sym.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "alpine-cross",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now we can take the x-derivative.  First, we now do need to specify 'x' as\n",
    "# a symbol.\n",
    "x = sym.Symbol('x')  \n",
    "z = sym.diff(f1(x), x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "random-scotland",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Of course, we can integrate too. \n",
    "z = sym.integrate(f1(x), x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "cordless-helena",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Here is another function:\n",
    "def g(x):\n",
    "    return x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "opponent-diabetes",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.91418507525795"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can nest the two functions:\n",
    "f(g(1.41))\n",
    "# which is equivalent to sin(1.41^2). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "interim-identifier",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "2*sin(x)*cos(x)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We should be able to get Python to do chain rule for us:\n",
    "z = sym.diff(g(f1(x)), x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "juvenile-sailing",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x}{2} - \\frac{\\sin{\\left(x \\right)} \\cos{\\left(x \\right)}}{2}$"
      ],
      "text/plain": [
       "x/2 - sin(x)*cos(x)/2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The integral:\n",
    "z = sym.integrate(g(f1(x)), x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "intensive-knitting",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin^{2}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)**2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We should be able to go back to sin^2x:\n",
    "sym.simplify(sym.diff(z, x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "hungry-toyota",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Everything above was for a function of a single variable.  You can, of\n",
    "# course, have a function of any number of variables.  Here's a simple\n",
    "# function of two variables:\n",
    "def k(x, y):\n",
    "    return x**2 + y**2\n",
    "k(1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "enormous-globe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x + 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "2*x + 2*sin(x)*cos(x)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Some more symbolic derivatives\n",
    "z = sym.diff(k(x, f1(x)), x)\n",
    "z"
   ]
  }
 ],
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