lektion7.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Computergestützte Mathematik zur Analysis\n",
    "\n",
    "29.11.2018"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "import numpy as np\n",
    "%matplotlib qt\n",
    "import matplotlib.pyplot as plt; "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "konkrete Polynome"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{x^{7}}{5040} + \\frac{x^{6}}{720} - \\frac{x^{5}}{120} + \\frac{x^{4}}{24} - \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - x + 1$$"
      ],
      "text/plain": [
       "    7      6     5    4    3    2        \n",
       "   x      x     x    x    x    x         \n",
       "- ──── + ─── - ─── + ── - ── + ── - x + 1\n",
       "  5040   720   120   24   6    2         "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Symbol('x')\n",
    "n = 8\n",
    "p = 0\n",
    "for j in range(n):\n",
    "    p += (-x)**j / factorial(j)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAI1JREFUKJHV0DEOQUEUheEvKFiEBXixBY2eHQiddbyFWAWJBYho7YGaFQjNlWiedycqJ5lMM/+dc39+zArPlvOA3gd0Rt0wcIIp9iUtTvHTPAuMA7iiC50EtI57I3ZqywD3eDzMVltGtV0WgGNAsyxQBXARAt75JqJYQB83hQIWUW2bBeCgUMBIg4A/yQsPQSSNTcSqPAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$7$$"
      ],
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "degree(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{1}{120}$$"
      ],
      "text/plain": [
       "-1/120"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coeff(x, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "xn = np.linspace(-2, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "wn = [p.subs(x, xx) for xx in xn]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "    plt.plot(xn, wn);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "aus `sympy` Funktionen können universale Funktionen generiert werden"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "fn = lambdify(x, p, 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(xn, fn(xn));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Grenzwerte"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "n = Symbol('n', integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{- \\sin{\\left (x \\right )} \\cos{\\left (x \\right )} + 1}{x^{2} - 1}$$"
      ],
      "text/plain": [
       "-sin(x)⋅cos(x) + 1\n",
       "──────────────────\n",
       "       2          \n",
       "      x  - 1      "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = (1-cos(x)*sin(x)) / (x**2-1)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAALCAYAAACZIGYHAAAABHNCSVQICAgIfAhkiAAAANNJREFUKJG10TFLglEUh/GfurU0SZvQGCI1OBcZ+BFc6hMI+jGaXFzEKQjaWtqiLRChpKU9hcBBMGgRRRBteN9BLu8LDXq289x7Hs79X/ZUN/jGAg84iPklPvCLZxTSBCX8oIYqhnhBEXN0cYF2LMwkSe7Q3OqLWGKCx+DuGypJkgFOA9bBCscBb6EB2eAgg03ARsghH/CNlOfco77VH2KMKV6DoXdcJUnO4oEaztFDXxT4XJRZGbf4TNsErkVfPMMTjmJewRfWouxO0gT/qTDH3dQfdTgpAZFH3n4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(f, x, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAAQCAYAAADwMZRfAAAABHNCSVQICAgIfAhkiAAAAQxJREFUOI3F07ErxVEcBfDPe29DiZJFymDQI4ZXNvKUP+Etz2ZTTHabyWKRSSlZlGyyKSlkUSahlKIoJT0peYZ7h1+/3uOnlDPdc7rndO+531vwM3owgw5cIY/FuH7J4JfHJS7whh0soI7eLAEwGA2dGMV95FtZA6ANpQRvRT9yvwn5X0zjVihyEy1Rn8AZnrHnm2KH8IQKpnCNfRRRwxrGsRIDG3azjvkEL+IdD9hO7T1GuVHIKYZT2io+0JfSlzFHGKYkcsIcJHGDArpSel2T62xgNsHbcYdHHKRMJ5hsFDISDRWM4RBHQuE1obMSlnDe7CRQFZ74FbvojnpZ+HSfQncDzQKyIN3j3+ALlbo2Fp9HJngAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\tilde{\\infty}$$"
      ],
      "text/plain": [
       "zoo"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tan(pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAALCAYAAAAa5F88AAAABHNCSVQICAgIfAhkiAAAAO5JREFUOI3d069Lw0EcxvHX1GQxDZtglCEalhUn+CesuL9A0D/DZLGIaTBYs9jEJoigYrFvuoGIoGCR6UDcwveEL4cTDYfgAxee53583tx9jn+sGrp4RROTIV/BFZ5xhJmUEPN4QhVraOMYJfSwj2XsBqhCKpA6tnK+hD4ecBCtPUdl1EEdDH4xmtH+SyxE2R7eMRvlO9j8NBPRZBtvoyi/0H3kCwEwrxuMo4jbXD6Q8Gka2Mj5KdzhESdR4QuspgJZDEWrWMIpzmRN3JP1UBnbuJbwRmBd9n1fcIjpkFfQwoesl+ZSQvxEY38N8K2Gk+85FrKxPB4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(tan(x), x, pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAALCAYAAACZIGYHAAAABHNCSVQICAgIfAhkiAAAANNJREFUKJG10TFLglEUh/GfurU0SZvQGCI1OBcZ+BFc6hMI+jGaXFzEKQjaWtqiLRChpKU9hcBBMGgRRRBteN9BLu8LDXq289x7Hs79X/ZUN/jGAg84iPklPvCLZxTSBCX8oIYqhnhBEXN0cYF2LMwkSe7Q3OqLWGKCx+DuGypJkgFOA9bBCscBb6EB2eAgg03ARsghH/CNlOfco77VH2KMKV6DoXdcJUnO4oEaztFDXxT4XJRZGbf4TNsErkVfPMMTjmJewRfWouxO0gT/qTDH3dQfdTgpAZFH3n4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(tan(x), x, pi/2, dir='-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "#limit(tan(x), x, pi/2, dir='+-')\n",
    "# ValueError"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$n^{2} \\left(- \\cos{\\left (\\pi \\left(n + \\frac{1}{n}\\right) \\right )} \\cos{\\left (\\pi \\left(3 n - \\frac{3}{n}\\right) \\right )} + 1\\right)$$"
      ],
      "text/plain": [
       " 2 ⎛     ⎛  ⎛    1⎞⎞    ⎛  ⎛      3⎞⎞    ⎞\n",
       "n ⋅⎜- cos⎜π⋅⎜n + ─⎟⎟⋅cos⎜π⋅⎜3⋅n - ─⎟⎟ + 1⎟\n",
       "   ⎝     ⎝  ⎝    n⎠⎠    ⎝  ⎝      n⎠⎠    ⎠"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = (1 - cos(pi*(n+1/n)) * cos(3*pi*(n-1/n))) * n**2\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAWCAYAAACVIF9YAAAABHNCSVQICAgIfAhkiAAAArxJREFUWIXt102IllUUB/DfaJEkOpaaQV+gaIQOtBAtNNHZiaS2CUGJFq4MxRaJwmxTcVEKLSJyChFcRJlBSBYxkkJU+LEwnBRTF1oaiF+jVjouzn3z4fY8876+rzoL/W8O99xz7/mf+5x7z3m4T/Ex/sGowSbSIF5BP95qxHgI/sR3JXNPoxuncA3HsRGP3QmWLWAIzmBXI8YzxGksz/QTROD9+BLr8X0aH8boO0S2WXTjb7TXM9wgSD+X6b9RHvh7Sf9h6xxbwoLEY1E9w17sz3Tj0+LfRToUMQKXcBnDW6bZPB5FH7YVlTnZFzAJOzJ9Z5K7cCObu4i9ycFLLRBcghO4gq1pP5iDX3AOO/Fsxfo+fIu5eLimzANcmGQe4PNJ/lax+ZEkJ1XSHxgd4rF6R6Tay9iOyfhaBLgw+f8CbRX77BB3cHaVox9xskT/kUjRpRXr3k3zawYMoxrdWFEYTxav9B/4rIRjp3I8gev4oKYofsEnMQ1fNUGwdqL9TayFKdhdGB/CZozBqsx2jziAMpwRBzC/pigGeAX/YmTJwvNJVj3BIzO720Wb/x/OMQzF2EzfrzpFa1wu1AbFAM+jB/PwULaoN8mqOzYxyao7Wg+/YmZh3I6V+EuUrWJAs8QXLsN4kQ35G/IflokTmp3pJ6hfJvo0XyZexFm8LgL4QbzMHWnfbkzFWhxU/QXfTjynVTl6SpSB90vmmin0n6a5N6scFrBYlIlLolMal/SdOJp4/SRKWRV6RBs5UAr7WeR/jrxVW+dWq9arvFXbkuaXDOSwQeSZk+Nx8YbU7ai6EqmOkrln8AlOi77vBDalzcuwX1z4e9GMvyF4z61nOCUZdrXocJSoSRta3KdRfC66qkcaMT4qUrUVvIqror7ebQwTdzdvCiqxGvsM/n9eo5iOA3htsIk8wAPcJm4C66ucp6goiiMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\langle 0, \\infty\\rangle$$"
      ],
      "text/plain": [
       "<0, ∞>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(a, n, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Das bedeutet:  Alle Häufungswerte sind $\\ge 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Man muss ein bisschen helfen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 4 n^{2} \\cos^{4}{\\left (\\frac{\\pi}{n} \\right )} + 3 n^{2} \\cos^{2}{\\left (\\frac{\\pi}{n} \\right )} + n^{2}$$"
      ],
      "text/plain": [
       "     2    4⎛π⎞      2    2⎛π⎞    2\n",
       "- 4⋅n ⋅cos ⎜─⎟ + 3⋅n ⋅cos ⎜─⎟ + n \n",
       "           ⎝n⎠            ⎝n⎠     "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 = a.expand(trig=True)\n",
    "a1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So wird die Annahme bzgl. $n$ genutzt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$5 \\pi^{2}$$"
      ],
      "text/plain": [
       "   2\n",
       "5⋅π "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(a1, n, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\left(\\frac{e}{n}\\right)^{n} n!}{\\sqrt{n}}$$"
      ],
      "text/plain": [
       "   n   \n",
       "⎛ℯ⎞    \n",
       "⎜─⎟ ⋅n!\n",
       "⎝n⎠    \n",
       "───────\n",
       "   √n  "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = factorial(n) * (exp(1)/n)**n / sqrt(n)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sqrt{2} \\sqrt{\\pi}$$"
      ],
      "text/plain": [
       "√2⋅√π"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(a, n, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Stirlingsche Formel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "y = Symbol('y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x*cos(x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x \\cos{\\left (x^{2} \\right )} - y \\cos{\\left (y^{2} \\right )}}{x - y}$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞        ⎛ 2⎞\n",
       "x⋅cos⎝x ⎠ - y⋅cos⎝y ⎠\n",
       "─────────────────────\n",
       "        x - y        "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = (f(x)-f(y)) / (x-y)\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANYAAAAaCAYAAADLyaLhAAAABHNCSVQICAgIfAhkiAAABmRJREFUeJztm2eMVUUYhh9gUSOo4KoRC7kGLEFjEEXCGhQxETG2WND4hxtRQiAaEEXBEIlG7KJisIAEjAUVu1gQdFcWNZRAQvGHigsEVkEpYlvZdf3xzQmHuTOnz5breZIN7JRv5ntn5kxdyMnJyQEmASuA34AdwAfAGa1aIzNdgBNT5I/q52kpyihnWlX/jikKbi0GAzOBKmAI0AgsBo5sxTrpdAQmAz+lsDGYaH4eBIxIUU45kuufAV2BJuDyhPkLQDMwN6P6gHztqjK0B8F+3g/0zbi89kxZ6V8J3Ay8A3wP/AXsAWqBkbibHXsgA+O8hPkLZDuwKoHPMrLlJ8jPHsCHDspsj5Sd/qNVwduAV4AHgTnAbhW+AOjgoNzXgdVAp4T5OyPr5B4Z1ecRYExGtvyE+bkQWb783yk7/Ycg06Q+Mx0LbEYG1zUZl/koUA/0zthuUiqQje7JGduN4uc44LWE9otI+wxOmL+t0F71T8xkpOFmaOETVPgES75TgQbgS0Pc48DPQB9L3iuAJYggDchMWkPp16xA6VLQH1YA5gO/AH8DK4HLLGWej5wimWbmpL6G+elxgSo7ycxdJP3AOhf5qm9F/KgHFgHDDWmHI37uQbYMa5F90cGGtFHbEez6u+xnHmn0T8ydiGPTtfABKvxNS75FyGnMmVr40wQ7O0rZrQdeAKYBs4HlyBGqnwL2gfUFsB34RtV9HjK4moALDeXeDXxtqVMSX8P89NNN2U+yiS6SbmDdgtS9AfHP03sNUK2lnabK2gE8i8wG61RYNbI094jTjmDX31U/85NG/0RUIF+kZmCoFtcZ+BNZKupcp/I8pYXPRL4MQ5BlpvfT1ZdmFdLIxxjsHqX9XsA+sJqBe7X0Q1X4Rwbbc7BvYuP6GsVPnUZlKy5Fkg+sPsA+YCdwuiH+BN//B6pyNiO+eFQg90TNyOrGI047gl1/V/1MJ6n+iXgMqfhCS3yNij/OF9YF2IJ8LY7Q0jdbfqb60qwC/gC6R6hfAfvAqsM8tW9CloY6C4FXA8qK42sUP3V2AWMD4m0UST6wZqi84yOknaXSjjLEnYKsBDb6wuK0IwTr76Kf6Vj1rwswaPp5OaAQgNtUum+xX+B6S4OrfWEPq7BiiH0bt7P/hHI6cBVwtCVtAfvAeteSpxbpBDpLkM5jw4WvfrYAd4SkqSNeG88NsbdCpYvyAmSVSms7BNik4rup3+O0IwTr71p78OlfoUX8gOwhorItIG4sMr1uAC5Clgomlql/BwBvIw00Hlkrz4tRFz9PIDPKGGRwj0MErEH2eysj2tltCW/EfC/XABwaYM+Fr366EN5+T7K/43r0Ba5UdajT4taE2PNsbY1QP29WqLfE1wM9VbrdxG/HIP1daw/R9E+FJ8BazOtjP92BfxGxQJ6NNAFnZVSXbsClyJesCfhVq1OB4FNBE9UqXuclZK9gw7Wv+4DrE+QrknwpmGTG6mWJ12csP2HtCMH6u9YekusfibsQcVZj3mCaWI+spW9UeWe6qRovUnqfViC7gTUFWSYG4crXw5W9fgnyFmmZPdZslXakIa43pXssG6Z2hHD9XfazNPqHMkUZX0m8R7HPq3x7kWPYtA9qL6F0mQv7T56G+cIKZDewLkY2sEFk7atHFfA7Zr/DKJLNqaDpWNp/KlilyvmRA/dKnZD9bDNwjy88TjtCuP6utAdN/ySNYGMEcB/y1VmKrIl16jB31mXISVFX5Mtn249FZT6y1q1VZXYABgH9keXI4pT2bSxFOslJSOcxkbWvHv2AT5H9X0uyAdkDPYesUt4DvkPe7J2DdGLvzu8r5MnRROTuagEygwxD/iSjFrnX8ojbjmH6u9IeHOo/lfATpmpL3kEqfjnZvCccjTwG3ojcX+xEGn0icJiWtkB2MxbIpedNAXXL2lePtyj9gkelSPIZy2OgqsN24B/kYOsT4FpD2huQwbIXGTjrkZnqEC1dnHb0CNLflfaQTn9nvI/MdP1buyIZcDzwcUC8C18rgc8ztNeeCdLfVT9rk/p7G8lnWrsiGfIApc9jwJ2vk0j+ZzPliEl/l/2szejfE3nXNQu5e1hH8P1Pe6Mz8BBy1+Xa117ArRnaKwc8/Vuin7Up/b3HlbuANzjwmUm5UIns1Vz7enbG9sqFSlqmn+X65+Tk5OTk5OTk5OTkOOY/l6RtQpn+45QAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- 2 y^{2} \\sin{\\left (y^{2} \\right )} + \\cos{\\left (y^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞      ⎛ 2⎞\n",
       "- 2⋅y ⋅sin⎝y ⎠ + cos⎝y ⎠"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.limit(x, y)"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Träger Operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\lim_{x \\to y^+}\\left(\\frac{x \\cos{\\left (x^{2} \\right )} - y \\cos{\\left (y^{2} \\right )}}{x - y}\\right)$$"
      ],
      "text/plain": [
       "     ⎛     ⎛ 2⎞        ⎛ 2⎞⎞\n",
       "     ⎜x⋅cos⎝x ⎠ - y⋅cos⎝y ⎠⎟\n",
       " lim ⎜─────────────────────⎟\n",
       "x─→y⁺⎝        x - y        ⎠"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l = Limit(b, x, y)\n",
    "l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 y^{2} \\sin{\\left (y^{2} \\right )} + \\cos{\\left (y^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞      ⎛ 2⎞\n",
       "- 2⋅y ⋅sin⎝y ⎠ + cos⎝y ⎠"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "if you really must"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import Markdown, display"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "Formel: $$\\lim_{x \\to y^+}\\left(\\frac{x \\cos{\\left (x^{2} \\right )} - y \\cos{\\left (y^{2} \\right )}}{x - y}\\right) = - 2 y^{2} \\sin{\\left (y^{2} \\right )} + \\cos{\\left (y^{2} \\right )}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Markdown(\"Formel: $${} = {}$$\".format(latex(l), latex(l.doit())))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Da ist aber der *fake math* Tür und Tor geöffnet."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\\lim_{x \\to y^+}\\left(\\frac{x \\cos{\\left (x^{2} \\right )} - y \\cos{\\left (y^{2} \\right )}}{x - y}\\right)\n"
     ]
    }
   ],
   "source": [
    "print(latex(l))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$3 e^{3 x}$$"
      ],
      "text/plain": [
       "   3⋅x\n",
       "3⋅ℯ   "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(3*x).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 x^{2} \\sin{\\left (x^{2} \\right )} + \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞      ⎛ 2⎞\n",
       "- 2⋅x ⋅sin⎝x ⎠ + cos⎝x ⎠"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = f(x).diff(x)\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 x \\left(2 x^{2} \\cos{\\left (x^{2} \\right )} + 3 \\sin{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "     ⎛   2    ⎛ 2⎞        ⎛ 2⎞⎞\n",
       "-2⋅x⋅⎝2⋅x ⋅cos⎝x ⎠ + 3⋅sin⎝x ⎠⎠"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ddf = f(x).diff(x, 2)\n",
    "ddf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 \\left(4 x^{4} \\sin{\\left (x^{2} \\right )} - 12 x^{2} \\cos{\\left (x^{2} \\right )} - 3 \\sin{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ⎛   4    ⎛ 2⎞       2    ⎛ 2⎞        ⎛ 2⎞⎞\n",
       "2⋅⎝4⋅x ⋅sin⎝x ⎠ - 12⋅x ⋅cos⎝x ⎠ - 3⋅sin⎝x ⎠⎠"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dddf = f(x).diff(x, 3)\n",
    "dddf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$x \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x⋅cos⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 x^{2} \\sin{\\left (x^{2} \\right )} + \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞      ⎛ 2⎞\n",
       "- 2⋅x ⋅sin⎝x ⎠ + cos⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 x \\left(2 x^{2} \\cos{\\left (x^{2} \\right )} + 3 \\sin{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "     ⎛   2    ⎛ 2⎞        ⎛ 2⎞⎞\n",
       "-2⋅x⋅⎝2⋅x ⋅cos⎝x ⎠ + 3⋅sin⎝x ⎠⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 \\left(4 x^{4} \\sin{\\left (x^{2} \\right )} - 12 x^{2} \\cos{\\left (x^{2} \\right )} - 3 \\sin{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ⎛   4    ⎛ 2⎞       2    ⎛ 2⎞        ⎛ 2⎞⎞\n",
       "2⋅⎝4⋅x ⋅sin⎝x ⎠ - 12⋅x ⋅cos⎝x ⎠ - 3⋅sin⎝x ⎠⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 x \\left(4 x^{4} \\cos{\\left (x^{2} \\right )} + 20 x^{2} \\sin{\\left (x^{2} \\right )} - 15 \\cos{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "    ⎛   4    ⎛ 2⎞       2    ⎛ 2⎞         ⎛ 2⎞⎞\n",
       "4⋅x⋅⎝4⋅x ⋅cos⎝x ⎠ + 20⋅x ⋅sin⎝x ⎠ - 15⋅cos⎝x ⎠⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 \\left(- 8 x^{6} \\sin{\\left (x^{2} \\right )} + 60 x^{4} \\cos{\\left (x^{2} \\right )} + 90 x^{2} \\sin{\\left (x^{2} \\right )} - 15 \\cos{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ⎛     6    ⎛ 2⎞       4    ⎛ 2⎞       2    ⎛ 2⎞         ⎛ 2⎞⎞\n",
       "4⋅⎝- 8⋅x ⋅sin⎝x ⎠ + 60⋅x ⋅cos⎝x ⎠ + 90⋅x ⋅sin⎝x ⎠ - 15⋅cos⎝x ⎠⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$8 x \\left(- 8 x^{6} \\cos{\\left (x^{2} \\right )} - 84 x^{4} \\sin{\\left (x^{2} \\right )} + 210 x^{2} \\cos{\\left (x^{2} \\right )} + 105 \\sin{\\left (x^{2} \\right )}\\right)$$"
      ],
      "text/plain": [
       "    ⎛     6    ⎛ 2⎞       4    ⎛ 2⎞        2    ⎛ 2⎞          ⎛ 2⎞⎞\n",
       "8⋅x⋅⎝- 8⋅x ⋅cos⎝x ⎠ - 84⋅x ⋅sin⎝x ⎠ + 210⋅x ⋅cos⎝x ⎠ + 105⋅sin⎝x ⎠⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for n in range(7):\n",
    "    display(f(x).diff(x, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Zusammenfassungen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "d6f = f(x).diff(x, 6)\n",
    "d6f += cos(x**2)*sin(x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(- 672 x^{5} + 840 x\\right) \\sin{\\left (x^{2} \\right )} + \\left(- 64 x^{7} + 1680 x^{3} + \\sin{\\left (x^{2} \\right )}\\right) \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "⎛       5        ⎞    ⎛ 2⎞   ⎛      7         3      ⎛ 2⎞⎞    ⎛ 2⎞\n",
       "⎝- 672⋅x  + 840⋅x⎠⋅sin⎝x ⎠ + ⎝- 64⋅x  + 1680⋅x  + sin⎝x ⎠⎠⋅cos⎝x ⎠"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d6f.expand().collect([cos(x**2), sin(x**2)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAAYCAYAAADXlTCJAAAABHNCSVQICAgIfAhkiAAACK5JREFUeJztm32QV1UZxz8gKisoROaoYVYYk2RhL0NpaohZoaOZJmlT420gx/Kl1FHDtH45CYSgkKaEmmvZGAphSUX0IoaNpqAWFpmxrAosii+rphu4tP3xfW6/+zu/+7737mp7vzN32N85z3nOuc99zvN2DlChQoUKAww1oMd5tvTngioMWEwHHgBeBLYCdwIHpR08uKRF/T/gUWCfwPPu/l1OhQGKicC1wKHAJKAb+C0wqh/X9LpHDXikvxdR4XWNHwJPA8MK5jsc2AEcF2h7P4oypxY8VyQOB5YAHcA2+3cFcIxD59EcCrvPDmfMG4FpwFLgn0AX8AJwD3rBIqKSGvAKsAnYANwKvLUAvnlxLJLfRvS+bcDtwCEpx3+eujynxdCNBn4AbEbfrR2YB7whz6IHMD4A/Ac4rwTe+6Dv+GGnfSnaZ8NLmLMBl9gCtgI3ATOAhSiPmO3QHow2U9jzO+OzzBlzhrVvBn4MzERK2Wnti4FBvXyHycBJKCz/qK2lg/4Ji76D3usZ4AZgFnrH7UiJPpcwfj8km5eI3+BjgKeM5g6b5/f2++/IsFZIhxVI5i0l8F4EPATs5LRPQN/q4hLm/B9Otkl+A+we0r9zBl73Gq/jnfZJKDxxPfXewBM25qQM86TBMKT8eS2yh9Y1MeO4vVEEswXYy+k70ni2xYwfhPK19cAVxG/wX1v/2U77lda+IMvCc8Ijn5xeSxiLDO/CEnhfgRzNARH964DHad78hWAwUraXgTf1ktdB6ENvJNtiL7ZxVzvtK6z9RKd9ENBqfbMSeN8FXJdhLUF45FPcD9q4n0X0v4g8cxS+gpTtCOonA2Eb/O3Wt4Fmw7k78C/0XYP5ZBEydeHR+w0+AXm5TTSmh1NCaKcAf0ApXhewFlWtdw2hPZ56JLcNRZB3A1926GbZOxwVsb68cpuLnMy4CL4A37TxH4+hyY3DjPntwBCUN16ElCxtrujjauP1rYzjLrBxVznt45En/BuNBmOu0SdZ26How34j43p8eORT3FHUlXRPp+8I47k0YuyBSGl9WdSI3uDTrO/7Ebx87x5U2t7KNAwevdvgX0RV5m1ID2egtOZhYKVDO4N6Knkd8o6PWNtKGqPN0629A72Xz/d+lHoGsdrWEFVcyyO375K8uUHpZA8wJ4EuF8415tcAf6G5WHY36Tx7C/A8EsJ+GeYfgixwlAVrtT7PfvvefhHNXmsO8BHgbciLLkPecv8M6wnCI7/ifhV54afRx58J3Ab8G3kDN3QHyWI1Ourz88Aa0RvcD9/Pj1jDNdb/Jae9lfQyTQOP/HIaB7wKPAe8K6R/dODvQ2yeJ1Aa5GMIOmd2c9k1yGiEyTpoeIehzb02Ya2tpJfbtUj3Jtla/SesmDbC+NyfMH8uzDTm3cBjyNoPR8JeTt0yJuE0wotrSZhj434R0T8aebR24CyjXQ7sEkL7ExSCbUeh3hKSrWccPHrnmU5Aihs0mI8Bn42gvwwZyGDkVCN6gy+M6QO43PqnO+1ZZJoGHvnl5Ed956agvd5oTw/pG4tkF6xtrEEpStJpwljjuyKBLovcok6XahG8u4i5lNUewzDsuSUwdjb1Y63xDt8W4EnrTwrX/2h0xyXQBXGOjVlHfKXbN0I9Ns9uGeZIi3ayybA1gd+FyGheiXLl3YD3UQ+b3ZOJCUbvttfIv8H9cPZrIX15ZdpOsXJ6wOjemWLuNUYbVax63PpH2u/z7PdmlPKcQHg06kcGi1KsoSxd3IS+P6CQJIj1KPRLi82Bv5+3f9uAPzt0XUghpyIFvDeC3zh0Y2cj8MuUazgTmI9ymqOQp4vC1sDfU9FZd9GYR10xfBwMfBK4GSl2EA/H8JqIjsmW0ljBfxD4FPAPFFYvQHIfAvzI2i/NsOYX7N8REf17OHRB5JVpkXIiwGtTirn99+yI6O8A3mJ0nci4PoMKauegtMlPOy9A6RBIz0E1mySUpYstgXUUihPRS7tFBx9+nhfmBXzMJz78cOELei3h+VEQp6JctsPG5K2I54FHvtDTTzvcoysfP7V+/1hwJOk94rwAnzxFNiheph75Q/Q8HnxMRL/rwYMYiS5sXY+i1Wep696+Nu6ehPnL0sXBxnd9QfwasCcqcnQSnkv8Cr3MKRHjhyLvm7a4dpHxe4jmCrOLY1A+vRaFVutsrWmUoQh45FNcP6+8LKJ/FY3pTAuq7oY9DxrtKvv9mQCfMda3gehjsldorAyXIVOPvsnBbzDasKudB9Ccg0fhRhoN7CBUDN0aOaJcXTzQ1rOkAF6huMUm+LbTfjSyLJ2EW0WoX6e8M8U8lxrtapJvlx2GlLMNXfMD+LSNvyPFXEXAI5/iTrFxW4A3O32TkUy7SHfLrEZ8np3loktZMvXIv8GDVfSwgmiwin4odYMWzKV3QuvvAb4eaP8Ezeks1CvukwNti4nO78vWxS8Yr7MK4BWKvVB1twddIJiDziO7kfBPjhnreqMonGZ03ajgUQt5PKMdj4xKB83hmB/SHZ4wXxHwyKe4g9GtwB50VHIzysl/jjZ3D7pnkAY14je4e1V1JvWrqo9SNyJlytQj/wYHnYPvQEdat6Hq/wLkCO5yaP0rwE8B30NFSf+YdRWNUWgnMrKLkU7PRUdRvpMJnpmfau1nOvP1hS7eivZFluPlzBiFrP4GFIo8i25ifShmjB9aPEnyzbUayfnlSmRBt6Di33tC+PiXAu5LfKPewyO/4u6Mag33oU3ejcLAZcDHMvCpEb/BQYpxE1LC7SgXnU89Sipbph692+CgSvYSJKPtqBC8HHlKF6egfPklVFz+K/LcbpHsDFTobEMe+DmUGl5I85XsXZCM/hRo6wtdHIGiub6KSitUGLCYjjbse/twzrPpu4i0QoUBjaEo+klTTyoCLShSWex2lPK/TipUGODoRqcWu6Ic/dWS53sHqjvMRnl+hQoVKlSoUKFChQoVXpv4LyjoEHe6JfmPAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 672 x^{5} + 840 x + \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "       5              ⎛ 2⎞\n",
       "- 672⋅x  + 840⋅x + cos⎝x ⎠"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d6f.expand().coeff(sin(x**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Partielle Ableitungen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{\\frac{x}{y}}$$"
      ],
      "text/plain": [
       " x\n",
       " ─\n",
       " y\n",
       "ℯ "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = exp(x/y)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAAyCAYAAAD/eF+lAAAABHNCSVQICAgIfAhkiAAAB+pJREFUeJztnXmwHVMexz95EpHxYoklYo0gYxBCJUoo5SWxJLZYixmjgsHUCKEUiqBKleIpKWIyxJbiUvZ9rSkEbxSGsiWMiS3EvoSIWCOR549vd91z+3bf29339u173/t9qrre7XP7nv5291l+53d+fR7ky87A1cBDwIE5a6nERGA2cD8wPmctRg+mDzAw5rFrA9Mz0PAQ8C1wTyB9E6AL+D8wDzgkZn5rAdfWS5yRitXzFpAlpwNrxDjur8BLwB4ZaBgLHEB5pRkC7OB9Xh/4GPhDjPw6gZ3qps5Iw1iKz67e3AFsnVHeVZlKeeF6DtjF+3wdcIbzXT/KC3a96IiR9+vApt7nKJ0XAnvVW5yRiunABnXOsw14pc55xmYo4SbMBGQuTQVu9NL2B2YBBeDQjPR0ULnSjALmI3MSwnUeB8wFrgFOzESlkYTBqEFLQ1SjONrJc0fgSec3+wE3pTxfLG7wThrGK2gssWqWAgJ0EF1p1kHjml0D6XnonIJ6vKXe9l/0sIIMQQ9wEfAz8AbpTNvZwIxUStNxDjLDlyLtDwPbVTh+GtANXBnx/c3AiBQ6whpFgPOAw7zPfYHPUEO6CvAyGgdnwrrAqxHfjQY+QIJrpQBcEPPYDsIrTX/gGeDoQHpeOicB+wJbAcOBi4DlwPbOMWsB76MCszOwOfLm/SmhrjbgC3RvaqFA/Ot7DDgWVZQRyBP5BTAo5Nhd0DOYR3Sl2R24L77UEsIaxaeRQ8rnWWBLZFl0pjxPLP5BuBdsI3QDhqHWZmSN5ylQW6XpA9wekkeeOsNYDPzd2b8YmRdRHA4sAzZz0v4JvAus56SNAb5BLeonyGnjMtLLZ5sq+gqkv7524DfkqHFZE1gAjEMFO6rS9AG+RtaCy8bArejeLQHuRc4en7BGcSDwn0A+lwN/Qz35mu4XbRGC0jIReCGQNgC4G3WH76MHf36dzxvFHO/c+6LCMcZL3w04AjgIjVXmotYvL51BVgH+jArW8076Qej+3gZ8hXSfTHE8dg96yOd5+2cARyKTZJGTzyTgUWAFMgNHB84/A7gema5ZMRCVv8WB9OvQdTxV5ffdqIHbx0kbhnqRj9Az7kDWzyzv+42QWToembl+ozge9TQuLwKXenq+i3E9qfmKxrjsCtTWgjeKAsl0jgB+QIX5W1TZXX7xtk40bjzWO36Kc8zeyKw7G40fRoWcZz7FuanTgfec7w5FBTnYgodRIP1zuBN4DTUQPiegQu+bTV1E9zSghu0qZ/9xL81lH3QvB6AGyB//HYx6IVClGhP43Y6oh+5X+TJqYxCq/e0Z5D0NFQ5/Ww78GkjbPYPzJqVWnasiO3oUqhhfUzpY/hX1DC6dlPcIz6OKNyHkHMORA8GfJNwVPbdBaJy3ADgto+vzmQ58jq7V54+oN3Qb3S4qV5qTKHq5NvOu46eAnp+BL6voeZnSygtyaE2q8rua2R61glkwCN1gf7sXmBlIG5DRuZNQb51zkDnh82FgH+TI+NHZH+ftryR8IvZM4BFnvz8av+yNeqe3iG5d63F9l6FCHBwvHYMK/Qpn6/auY4WnM8hfgLe9zweiMcyWIVsSz9cWXp7/ijqgb4LMqtFOdKXpTphXn8D+Ykpt3++9/feIJuk50+JqTaOzEm3Aas7+c6hFdhmOKhNopvw+1AIfjDxwEwPHT0Jmlc8yZCYdAExGBXF5hJ5ar28mGkuOpbx3fAC1+C43IhPpYtSjBVlC0bJZ7n3+0tOVlgWU3+PM2I3kYtMGbBbIZkxT78DMAvF1XoJMm6FobNOJWlm30I9GheNc1IIejgapU5B58qn3HajnX0mpubSe9/vBgXPP8I59LKZWnwLxr28WGmONQ7P5/lbJnO+isnk2Hs2ngNzFi4AH0XhkCxTBcRXljXDTMBK17kHbMA5JAzYLZOsIqFdgZoH4Oguox1iGHCpzKPUM+eyHvEa/AO8gb986aHAf1HwXmm/wOY5Sb5zPZGQCbRtTq6v5gpjHdkdslX7fReVKsz/qiXxGIa/bEtSAz0UmZ9OyProJcbwuLlkGbKalpwZmPgicFZL+BKVeqFZhMuXzKy3HEsJDI6rF8mQZsJlUS08OzDyL4qC4DZlp05Ana+2oHzUx56D5pJbm32juIEhULE8jAjbjaoHeFZjZgcYx8ymPvWsVbkGz9i3NVFTgwmhYLE8MmkmLkZ63KA0ZakkGI1dimLciMpYnB5pJi5GOoYQ7NVqSKwgfExyBggRPaaycUJpJi5GOc1FcXY9gXUonz3waEssTk2bSYiSnHwo4TTO90bQcD+wZSGtILE9MmkmLkZyTyNF5Ue9XA3xmo8C7doqxPD+ieYI8aSYtRjq2RVEQPWY849JG+Ft5hlELQ/IWEGQh0eEOYdstuag0jBwJRjkvIFl4/2fVDzEMwzB6NfUKmW7UuyuGkTtN+56BEZuF6D2Vlehd+LG5qjGMFmAh2azLYESQ1TyNYfRYrNK0Pt3oRayXgKNy1mIYLcGG3t8hwJukW9/YSECPCnjrpfiLmfyA1nbui16kM4xeSbV1llen+N/m2tHqlMElZg2jV3E3Wqjd5WmKK7QMQ73KPOB/wKmNk2YYzUkt6ywbRq8kyTrLhmGQbJ1lwzA8XkCLcS9FS14ZhlGFtOssG0avJe06y4bRa2nVdZYNo6H0hHWWDaOhdND66ywbhmEYhmEYhmEYTcjvnirvfT0VIOEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{\\left(\\frac{x^{3}}{y^{3}} + \\frac{12 x^{2}}{y^{2}} + \\frac{36 x}{y} + 24\\right) e^{\\frac{x}{y}}}{y^{5}}$$"
      ],
      "text/plain": [
       "                           x \n",
       " ⎛ 3       2            ⎞  ─ \n",
       " ⎜x    12⋅x    36⋅x     ⎟  y \n",
       "-⎜── + ───── + ──── + 24⎟⋅ℯ  \n",
       " ⎜ 3      2     y       ⎟    \n",
       " ⎝y      y              ⎠    \n",
       "─────────────────────────────\n",
       "               5             \n",
       "              y              "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x,2, y,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAApCAYAAADzhF+CAAAABHNCSVQICAgIfAhkiAAACRJJREFUeJztnX2wF1UZxz/34sslbgKXUG+jIZpmCEiFEebLHWLIlxgxtXJ6IbXSJqzGMExnelNCcwgry3F0HGlSk/4wSit6sSuF5tCbGRAZcSmTMsCXJJHQ2x/fs3P37t397dnf7v7O/rbzmdnZu7vnd87z7Nmz+zzPebng8Xg8jnkQeAT4A/Bpx7KUzeFAP7AR6fx2p9J4/q850UGZB5n9KOAhYIYDGVpFL3C8+ftg4G/Ay9yJ47FgpmsBstJpkWY+8MayBYnhWbM/wGx1Zjv6ogM8CTwFvMKdOB4L3gVMcS1EkUwF7nZY/sPAv4HrHJV/CvA94AlgEDg3Js2ngPXo5fQvk35qjjJnApuAjhx55OEjwO+RPs8iq+rMmHS9wEqk8/PAo8CpLZIxK1nr6EpU3zc2SDMGWMOQBeqCW4EVRWS0P/AbYFIRmeXgIHRT8zSgZjkduAb50EmNfQ1wAZJvGnAP8A+gp4nyJiC/3YXbFHAWcAZwNHAMsBT4LzA9lGYc8BfgG8jqmwy8BXhtSyW1J0sdvQnYiiytRo0ddK/uKE7MTHQiHfqKyOwK4GtFZBTDecALDH+RfBl4DJgYk34JsLgkWWxJauxRuoEXkfsTYKPvgcBa4L25JS2eXcDFoeMvAOsapM9av60mro4AxgJbgDkoYJrW2AF+DswtUjjs7t9sYCewnzl+HLgsks8Mk09Dd2Ms8DRwVC6Rk+kAfgXcYo4XA/8MlTeWIaW6gJ8CbytJFltsG3uvSfvm0Lk0fTuAu4DPFiFogYwCzgf2oi9iwEZgOXAnijH8DljEkOuRpq9r4uoI5LIGLmM/do19PuoxKtLtsrl/1yLLKuDb6BkK8zMsdFgC3N+spJbMQ+bhFciXCkc3jwB+jXzHqnS92Tb2u4HfooYSppG+JwEvoUYTbNNwxzTgOWAfChaeEbm+x2zLgNchE/k55O8HNNLXNXF19EH0zAXB4H7sGnsn8Hdk0hdJ2v3bxPAu2suAP4eOz0EW2YS0grYCl+SR1JIH0QN1WgvKyotNY78eRdZfnXC9XfQ9AOkwEzXoHQyPmexFgbswy9AXP0wV9Y2ro9egwN2xoXP92DV2kJXTX4BsUZLu3zEoKDomdO5E9Iz2IJdwC/DxtAJmmx+VHZibA+xGX7TXl1xWEaQ19uXI1Eryj9pN3zA/QZHfgG2RY1CsYXfouIr6JtXR+1H97gttg0j2fajxNGKuSfuqAmVtdP8uB+6NnDsQ+efzkDXwRxRkb8j1wEBOQdM4HsUEFgLfAX5QcnlF0Kixf4XGDb0d9Q1zP/DN0PGdKDAV5mqGvuxV1LdRHY1Dlkt4W4/0nEq6Pz4auTXRAFmUc9C92IGso8dQN1/U5Uu7f78APhCT/y+BryKz3yrG9bApoCwmIR/nKnM8Hb29Ti6xzGbpRhHNGaixLzZ/h9/gX0c3dw5waGjrNtfbSV9Q4OdkFDeZhszzl1A3ZMAJyJ+8CpnD5wHPIJ+9ivqm1VEc/dib8aBu6u8mXBuFgmeDqIHfBNwAbDbnVobSpt2/iejeHxJTzgqTdo2NwF0mo6ttEjdBDwos3Bw5vwq9rapGH6qM6HZ7KE3c9UEUWW83fUG6bUMm4ZPIhH9rTLozUV/0HuBPwEdRMKiK+jaqoyT6ydbYV6KAWBw3mvKWMdRVBjKz15lrU7B7Xi5EvnwcC5HbcZyNwFNNwRenJfR4PMP4DGo7h0bOz0Jf2yRr+UPmdxdYlrMa+GTCtR/TYGzMfpHjoB/viZQCB8gWwLsDeE+G9HkYoLqylcEA9dN3gPbTabvZH4lGtgVcinz+/xBvSQS9HLb99OsY3p/eiUz7i0xe70j6YbSx95r90ykFbkHmmy1pL48iqbJsZVBHfdtRpx1m3xs5P8/sz0/5/V8ty/li5PgUFEDdjAKAT1nmwyeQSXGC7Q88Hg+ggUeDwPtC57rMuQecSBQh+mUP5lBneatmZbCAPFoxI6wIOYukjjq3o05JMgdtJrwOQZC2EtOVo/PZ95p9amd8Djost0XABrPdhl5MwbVWYCtn2jYdRa2DbQ9wdhP5tII8ek5GY7E3of72iW2o0zY0VfcRFI3PInPQZl4InXseDfmeQvLqQycxsp+9JSxCb0KXUyxBb8ItyAzqQPOPXU+EKYJu5NuNSUvYhjyA/EfQRKa0UWdVZIDGfe+NWIDaTjRAFoxxH0TR8uWoP3wVesZtffXcRM34nWY/rlUCJNCJZOtCUxFHo9FP7c5ZqN96d1rCNuM49ECvNcfPOJTFFcELfGfk/I/Qx3MJ+or3ofuzHT0Lq1ok3whmoTfQRa4ECHEpGvW0k2yDG6rMamTC140FaEGI1Wg22efditM0W9HMt/XAuzP+9nLUdiYXLVRZ9CCBP+dYjvHAD5E8o5H/1OdQniIYi0akdbkWpATORV0+k5Dveh/t+VJ7pdn3olhRlmnGNyB/3Yn/bUM0QLcLjd21Gm6Xg7RVNeYif2YXCnLch5tFL22wXSFkAXqBldnTURZpOj6OvojbkDl/L9VbDdimnoL++u3A94E3ZMj/WLTgxIs5ZCyVuNVl11J+RT3EyL78FWhljo1oKeXZ6Cs4Cn3VN5csU7Ok6RLwTuBbrRKqYNJ0XI/GxU9AAdVTGTm33TVpOowBXm7Od6NJMxsy5D+DkTMBK898ZMofXmIZNqtqLEXdOBvQ9ERXq62mYaNLDxpCWWaXZpnY6DgPdTM9imIsVauvNB2ORCsEBf+Y5GMZ8j4CtZmqzmRMZH/UPXRhiWU0tapGRamTLknUQccydfgwskar9oKz4lrUZVAWTa2qUVHqpEsSddCxTB3WUN608NI5BP1zhjJN+cyralSYOumSRB10LEOHw9BimwcXlJ8TrqGg/zaRQKZVNSpOnXRJog46lqHDUuBLBebnhNEo2HJYSflnWlWj4tRJlyTqoGPROoxH7oDLfwFVGLMYvhB9kTRcVaPNqJMuSdRBx6J1uIlqLZOdm0vQf6wsgk4UD7gSDVwYX1C+LqiTLknUQceydJiP1n+oHQsLyqcP+UybcD+zLi991EeXJPpofx37KEcHv06jx+PxeDwej8fj8Xg8Ho/H4/F4PJ4I/wNGGmsH5DrMHwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(- \\frac{x^{3}}{y^{8}} - \\frac{12 x^{2}}{y^{7}} - \\frac{36 x}{y^{6}} - \\frac{24}{y^{5}}\\right) e^{\\frac{x}{y}}$$"
      ],
      "text/plain": [
       "                            x\n",
       "⎛   3       2            ⎞  ─\n",
       "⎜  x    12⋅x    36⋅x   24⎟  y\n",
       "⎜- ── - ───── - ──── - ──⎟⋅ℯ \n",
       "⎜   8      7      6     5⎟   \n",
       "⎝  y      y      y     y ⎠   "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x,2,y,3).expand().collect(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAAWCAYAAACcy/8iAAAABHNCSVQICAgIfAhkiAAAApVJREFUWIXt112Iz2kUB/DPLDYvd6JNEbVWQ15qIylKu1ove6EGW65YiYu1N0ITN240cyNkS14uhra21V642FrtSikvW9wgtIkQsfKWWBnsuDjP5Ofx+88M/n9D5lu/+p1znnOe7/P8nnPO86MHHyR2YVN6/wGncD89x/BtN/GqCT7BDUxP8lzMwRcYhQ14gvHdQe51sQCPMbyg24LzGJzkKbiN3h3EuYPltSBYbdThBHYmeRX+xeeFMc3YU8G/FxaiFeNqxLHq+EYcyUaRkxMz+zk0ZLpxeICnuCuOeLdgPdo6eaaX+B0V5Gdl+lF4hAGZ/lOMFJvThFsYW0bozzRpvmN1aEm25hK/efgjBW4VObZWHKkiBqG+k6d/5vMVHuJ/fJnZVuP3soVkOCAq+SuYgGc4m5HdKBa7IxvfC78k23lsw2b8k3S7u0CmI0zAPSzCPrGpRRzG0i7EOYifKxlbBNnFSV6b5F9FCyjip2Rr8nKV7IMjyTamC4TKMBzXsC7J48VXnpbkwSK3P8v8mtOYESKXm5Lf7EoTDRV5cQkrEun9Ii+KmJwC7asQZ1ny/b6DRVXCQFGMtmf6veKrwhKR2zlacFm0s5viOM9sN5b1rqviWDZiawraIHKziB9Fbv8nilGO9iJRV2LrDHcwukT/XeF9rvLNXvwG81npRdWsrzDmps4rbhtmvAmBLmANhlUj0EJxVK8LwttKxvRNtkPVmLA7MUcc3dOiKJwThSH/yv3Egs+8U3ZVxlSRjxcxJOnmi4WV5cpJ5T27GC/vw+8N2vvddS/fU+G4WNi0TN9+5WvDX6JXbxKV9AKu1JDvW2Gk+L26q/wXaoZY1N8ltkn4Lfk/ETet06KdfF0Lsj3oQQ8+bjwHtYGke7oCquoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x e^{- \\frac{x}{3}}$$"
      ],
      "text/plain": [
       "   -x \n",
       "   ───\n",
       "    3 \n",
       "x⋅ℯ   "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1 = x*exp(-x/3)\n",
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int x e^{- \\frac{x}{3}}\\, dx$$"
      ],
      "text/plain": [
       "⌠          \n",
       "⎮    -x    \n",
       "⎮    ───   \n",
       "⎮     3    \n",
       "⎮ x⋅ℯ    dx\n",
       "⌡          "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1 = Integral(f1, x)  # träger Operator\n",
    "I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(- 3 x - 9\\right) e^{- \\frac{x}{3}}$$"
      ],
      "text/plain": [
       "            -x \n",
       "            ───\n",
       "             3 \n",
       "(-3⋅x - 9)⋅ℯ   "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1 = I1.doit()\n",
    "F1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1.diff(x) == f1     "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ist aber doch nicht falsch."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAM5JREFUOI3N0rEyQ0EUxvEfJcMVWhoZ8Qpq75PnoDJ0Kl5BK0U67yAUiUYmpZGYoRDNuePOkXszYxS+5ps95/x3Z3c//lC7uMIz3jHCOVpNUBsTzHGDU/RjfY+dOvA2hrqpfhb1y0XQfjSHWE29DUwxw7o0cBzew2cCX3GHNRxl8DD8oeYaj+GdDBbhLzVgWd/K4DKthM8zWO5YWKzN6lwVHIR3asCD8B9v0Lb8O97Ed2T9KgDlqdXInfiO3EBD5GAP1xjjA0+4wHYT9I/1BRgnMa7SDNsAAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F1.diff(x) - f1).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int_{0}^{\\infty} x e^{- \\frac{x}{3}}\\, dx$$"
      ],
      "text/plain": [
       "∞          \n",
       "⌠          \n",
       "⎮    -x    \n",
       "⎮    ───   \n",
       "⎮     3    \n",
       "⎮ x⋅ℯ    dx\n",
       "⌡          \n",
       "0          "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1b = Integral(f1, (x, 0, oo))\n",
    "I1b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAPtJREFUOI3N0r8rhVEYB/CP61UskgxGTIpB5F9QTDIqpbsqi/wfbJLJKLJJ+QNYxHAt0rVSkqgrGa7hfa5er+NaDL7LOX1/nPOc5zn8ETpQxRle0MAFVtHZLriLJu6xg01cBbcfB3/DfBjqGCjwXTgMbbndbSsJbTy08xZRKYiDsdYTwRY3ib5y8CHW4URwpLAfLYuLUc4N+gt8hoPQmpgtBys4CvEO29hADa+4Dm0mUZEMa7gM8zOOMYXTCE6kgj+hJw5qyMfzpTntsIRu7OE9ZehNcNN4lH/Bz+5mJdNJlFQL4xjm8IYF6RmDdfnveArzLbYw9Otj/i8+ALxbOoNSpZ4PAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$9$$"
      ],
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1b.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAPtJREFUOI3N0r8rhVEYB/CP61UskgxGTIpB5F9QTDIqpbsqi/wfbJLJKLJJ+QNYxHAt0rVSkqgrGa7hfa5er+NaDL7LOX1/nPOc5zn8ETpQxRle0MAFVtHZLriLJu6xg01cBbcfB3/DfBjqGCjwXTgMbbndbSsJbTy08xZRKYiDsdYTwRY3ib5y8CHW4URwpLAfLYuLUc4N+gt8hoPQmpgtBys4CvEO29hADa+4Dm0mUZEMa7gM8zOOMYXTCE6kgj+hJw5qyMfzpTntsIRu7OE9ZehNcNN4lH/Bz+5mJdNJlFQL4xjm8IYF6RmDdfnveArzLbYw9Otj/i8+ALxbOoNSpZ4PAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$9$$"
      ],
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1.limit(x, oo) - F1.subs(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "n = Symbol('n')\n",
    "f2 = x**n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int x^{n}\\, dx$$"
      ],
      "text/plain": [
       "⌠      \n",
       "⎮  n   \n",
       "⎮ x  dx\n",
       "⌡      "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2 = Integral(f2, x)\n",
    "I2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} \\frac{x^{n + 1}}{n + 1} & \\text{for}\\: n \\neq -1 \\\\\\log{\\left (x \\right )} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧ n + 1            \n",
       "⎪x                 \n",
       "⎪──────  for n ≠ -1\n",
       "⎨n + 1             \n",
       "⎪                  \n",
       "⎪log(x)  otherwise \n",
       "⎩                  "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "m = Symbol('m', zero=False)\n",
    "n = m-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int x^{m - 1}\\, dx$$"
      ],
      "text/plain": [
       "⌠          \n",
       "⎮  m - 1   \n",
       "⎮ x      dx\n",
       "⌡          "
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f3 = x**n\n",
    "I3 = Integral(f3, x)\n",
    "I3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x^{m}}{m}$$"
      ],
      "text/plain": [
       " m\n",
       "x \n",
       "──\n",
       "m "
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I3.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x^{n + 1}}{n + 1}$$"
      ],
      "text/plain": [
       " n + 1\n",
       "x     \n",
       "──────\n",
       "n + 1 "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = Symbol('n')\n",
    "I3.doit().subs(m, n+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Einfacher, aber gefährlich"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x^{n + 1}}{n + 1}$$"
      ],
      "text/plain": [
       " n + 1\n",
       "x     \n",
       "──────\n",
       "n + 1 "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2.doit(conds=\"none\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sqrt{e^{x} + 4}$$"
      ],
      "text/plain": [
       "   ________\n",
       "  ╱  x     \n",
       "╲╱  ℯ  + 4 "
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f4 = sqrt(exp(x)+4)\n",
    "f4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int_{0}^{1} \\sqrt{e^{x} + 4}\\, dx$$"
      ],
      "text/plain": [
       "1               \n",
       "⌠               \n",
       "⎮    ________   \n",
       "⎮   ╱  x        \n",
       "⎮ ╲╱  ℯ  + 4  dx\n",
       "⌡               \n",
       "0               "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4 = Integral(f4, (x,0,1))\n",
    "I4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2cAAAAfCAYAAABj0rVUAAAABHNCSVQICAgIfAhkiAAAC5lJREFUeJztnXusHUUdxz9tbyk+aoutWrXgQahGxaitom2qvYEAQYImYBETlSMSEjVBjSkqWnM1II0YRaVIUWONDZhog4QIMRKhgvCHPKoIWhV7bLAtUh9YK/SB1z9+e3LPPXf3nNnd2TP7+H6Sk5u7z3l8vzOzM7OzIIQQQgghhBBCiFIyJ3QAhBPKp/woDZuB8jk/SsNiULrmR2k4E6WJKJT5wKTnn0jmFcCZoQPhgSJ0U8bfMl8J1kDqonUfNMEv8kp25JUpivCKtJmdumhTunJjPnAa4euTon+l513AGaED0RCOAa4HZoUOiAeaoJtZwCZgYeiAVJA6ad0HdfeLvJIdeWU6vr0ibWanTtqUrtyoe11VGa4BjgodiIbwVWB56EB4oim6WQF8JXQgKkidtO6DJvhFXsmGvDKdIrwibWajTtqUrtxoQl3F7NABGMKc6HcodEAawLHAKuCB0AHxQJN0cz+wGjgudEAqRJ207oOm+EVeSY+8Mp2ivCJtpqdO2pSu3GhKXVX6h7NVwD2hA9EQ1gFbQwfCE03TzVbg0tCBqBB10roPmuQXeSUd8sp0ivSKtJmOOmlTunKjSXVVqbkSWBQ6EA1gNvA3YE3ogHiiaboZB55AqzO5UDet+6BJfhlHXnFFXplJkV4ZR9p0pW7alK7caFJdVWquDx2AEdHCVmfZHOj+b4juvyDQ/X3TFN10WYjlXxXm3reQ1stGk/xSJa9AWL/IKzMp0itV0mYLleM+ka7caExd5XNa4yLgIuAm4E/AU8CTwN3ABzPcaxmwY8D+DslLUO5NeS/f+E6LolkBPI6FsZ+qxWWYbqrA+5jS8kUOx/8Ly78VRQYqhqppAwZrPS9VTI+q+6UqXoHq6aNIr0D10qPoNom06U6T2iwdpCuofl0FKeqrMY83XQt8E9gD3AHsAl4EnAN8G/sOxVrc1+9/B3DLkGOeBK6O2f4fx3sUhe+0KJoTSK6AqxYXF92UmWOBb2Aafm6K8/4NHF9IiJKpmjZgsNbzUsX0qLJfquQVqJ4+ivQKVC89RtEmkTbdaFqbRbqqdl0F2eur3JwCnM3MJ+ElWIZMAuemuN6mIfs70a+MpE2LFmGnCFwH/Cphn+987dKOzh3PcO4ghummCNr4icss4HbgUeAq3EcDAO4Drs143zbZwp9FGy3Kq/W8FOUVkF/6CeWVPFTNL0V6Barnl1G0SUKU46A2iyttpKs0+MyLqtZVkKG+8jm8+3PsqfZ/fdv3YkaCmRGcDVzOzK+7Px/4u8ewjZosaTGI84BfYL0nTwEPAZ8G5sUcOwv4KPAI8DTwV+y7EAtINvnc6Ng4fMfFBz50cy5wG7APW5b1j8BlhH1p9hKsMPsAcCDluU8z+m9/FKGNkFrPSxm9Avn9Iq/4oU71gg/K6JfQbRJpU22WIqiKrtS2i/A5rXEQh6O/R/q2nw6sBF6CJWaXtwO3Olx3HvBe7BsOB4DfYAXCM3kCWzBJaZHEF7FCbR9wAzYkema0/QzgtJ5rAmwEPgTsxl6ePIQNB5+MFWi9x3b5LzA/TSQi0sbFF3l0MwfYApyPzYH+IXAQS9MrgFcCF3gOrwuvAjYAX8M0fErK8+dj+VgWsmijzFrPSyivQHa/yCujo071gg+qWLdA/jaJtKk2SxxN0ZXadiNkDOs1mcSM2c/rsSVRe0fxNjJ8VK9D/AuSf6a8y6smpUWL+CkCK6Ptu7Bh4N7r3BLtu6xn+1ujbTuwFXq6HIWJYpL4XqjPYumWhmH5Oow2+YaLs+rmmui+VzK9c2Iu8Mto36tThqVNvriMYdMOdgDPirZNkG6q1k6sQsxCG79TNQZpo0X1tJ6XvF6BMH4pwiuQLy6hvVIEaf1SZ69AeL+EbJOUqRwHtVn6aSNd+WBQXqhtx2hWrdkAnIQ99f40Zv927Il+VfT/XKw3oH8YtJ/vAqdiBcBzgNdic1Jb2BP363KGuwiGpUU/F0Z/L2f6qjxHgE9gadSbwd1egSuwFXq6HGKwMXcAix3C00vauPgmi27eDHwYuBlLj94em8PA93qOGyWfw5YGbmNTQLKwCPiDrwDlJIs2yqz1vIT2CqT3i7wyOupUL/ggtF9CtkmkTUNtlimapiu17WLokLxkZ9xvy5DrXRId9ztszmgSX8dekgMb8s76sibAl6N73pTinA5+4x3HoLRoEd8LdX+0/cSEa/4l2r+w7/iXxxw7BxNpJ2bf0ui8VmLop+Oar106pEvfzY7hSKubLdH1b8B6L/p/P4r2XxhzblFxORkrSL7Ut30C99GA46NjX+xwbCdF2NPkRZdh2mglXLcsWu8QpgzME4bNjtdN4xcfXvEdl7J7xXe9APF+qatXoDx+CdEmKVM5Ds1us4B0FUJXXRrftut/5+xR0r0AvHvAvo9gcywfwZ76/zHg2Jux5TbXYXNOP58iDP1ch/XQvC3FOT7jHUeatOil+4HFPQn792BzkBdgvU7d4x+POfYZkl+ofAzrrVjO8FWBssTlaqZPWQAbun4n1qPRf8/tDteE9Lo5Pfr7niHH7Rqwz2dcxoDvYz1a64eEaRDLsZWrknTSS1F5Adl1DuXReqgysJcy+MWHV8BfXEJ4RfXC6LwC5fJLiDZJWcpxqL42e5Guqqcrte0K4mPYk+FDwAsdjh8D/om9OJd3iePnRfcucpWpNLikRYvBvVAnJJzX3wv1ANl6ocCGhIelfdp8HUQ7utZ4jmuk0c3R0f225bhfEm2yxWUh7r00cd846XItcHHKe/fSJn9euGqjRTW0nhefXoHR+qVIr0C2uJTFK77I45e6eQXK55cQbZIylOOgNssg2khXWUmTF41v2xWxWuMnsfmk27HhyH0O5xzB5p6uB+7Kef+V0d8QLzL3kyUtenkQ6/UYx3opezkRG9rfydRc7Qex+a2rmRn/tzA4v78F3IutCnQwZn/euBRBGt3Miv6GeIciiYPAdxL2Lcfy8m5sfv29CcfNA96ITRUIhQ9tlEnreSmjV8DdL/JKsdSpXvBBGf0y6jaJtKk2S1rqrCu17TyzHnsSvA/3eb1dzovOXepw7GsSrv8y7JsGk0xfESgEadKiRXwv1Kpo+07gBT3b5wA/jvZ9pmf7mmjbDqamC4CtfLQt2tcZEI7ziZ+Tmydfk2jjpzcmjW5+HR17TsL+1WT7FkYbP3HpZQKHeclYfr07573aZA9/Wm20KLfW81KEV2D0finKK+DfLxOMzit58eGXungFyu2XUbZJQpfjoDaLC22kq7RkzYtGt+18jpxdAHwBmyd8F/FP6h2SX6K7DXuCfMzhXmuBTwF3YIXAfmwY/SxsiPNW7GXJUORNiy73YC8TXgr8Fnup8QD23YaTsCfvq3qO34Z9J+Ri4GFgKzYt4GzsY5C7GbzizQ+wVZYWM9Wz4SsuRZFGN+uAn2Dpcjv2rZDZwEuBFdiqQMcVE8xCWIy97BtqTrNPbZRB63kpu1fA3S/yin/qVC/4oOx+GVWbRNpUmyWJJuuqyW07r0wwfG7lnUOucYzjvdYANwK/x4bHDwNPAD8D3s/UMGcoJkiXFi3ie6G6nI8Vavux+cUPY71PR8ccOxv4OJY2B7HCbSPWK7Wf4S9wjjE1XJ4lLq608dcj4aobgDdhFcZeTDf7sDnQm7AXVLPQJkzvykr8dLC0yRb+CdJro0V5tZ6XCYrxCoTxSxFegTAjZ768kocJ/Pqlyl6BavhlFG2S0OU4qM3iShvpKg0T5MuLprbtRINYhgnixtABEaJgpHUh3JBXRFmRNoUQtWEJMz8u/mxsyHcSm8crRB2Q1oVwQ14RZUXaFKKP0NP/hH82YN96uBP7rsUSbEh3KTaH9yyswBOi6kjrQrghr4iyIm0KIWrPqViBtgebv30AW652HfZSpBB1QVoXwg15RZQVaVMIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQIoH/A1sFAHdADyugAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 2 \\sqrt{5} - 2 \\log{\\left (2 + \\sqrt{e + 4} \\right )} + 2 \\log{\\left (-2 + \\sqrt{e + 4} \\right )} - 2 \\log{\\left (-2 + \\sqrt{5} \\right )} + 2 \\log{\\left (2 + \\sqrt{5} \\right )} + 2 \\sqrt{e + 4}$$"
      ],
      "text/plain": [
       "             ⎛      _______⎞        ⎛       _______⎞                          \n",
       "-2⋅√5 - 2⋅log⎝2 + ╲╱ ℯ + 4 ⎠ + 2⋅log⎝-2 + ╲╱ ℯ + 4 ⎠ - 2⋅log(-2 + √5) + 2⋅log(\n",
       "\n",
       "              _______\n",
       "2 + √5) + 2⋅╲╱ ℯ + 4 "
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Das konnte sympy vor zwei Jahren noch nicht.  Probe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 \\sqrt{e^{x} + 4} + 2 \\log{\\left (\\sqrt{e^{x} + 4} - 2 \\right )} - 2 \\log{\\left (\\sqrt{e^{x} + 4} + 2 \\right )}$$"
      ],
      "text/plain": [
       "     ________        ⎛   ________    ⎞        ⎛   ________    ⎞\n",
       "    ╱  x             ⎜  ╱  x         ⎟        ⎜  ╱  x         ⎟\n",
       "2⋅╲╱  ℯ  + 4  + 2⋅log⎝╲╱  ℯ  + 4  - 2⎠ - 2⋅log⎝╲╱  ℯ  + 4  + 2⎠"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F4 = Integral(f4, x).doit()\n",
    "F4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 \\sqrt{5} - 2 \\log{\\left (2 + \\sqrt{e + 4} \\right )} + 2 \\log{\\left (-2 + \\sqrt{e + 4} \\right )} - 2 \\log{\\left (-2 + \\sqrt{5} \\right )} + 2 \\log{\\left (2 + \\sqrt{5} \\right )} + 2 \\sqrt{e + 4}$$"
      ],
      "text/plain": [
       "             ⎛      _______⎞        ⎛       _______⎞                          \n",
       "-2⋅√5 - 2⋅log⎝2 + ╲╱ ℯ + 4 ⎠ + 2⋅log⎝-2 + ╲╱ ℯ + 4 ⎠ - 2⋅log(-2 + √5) + 2⋅log(\n",
       "\n",
       "              _______\n",
       "2 + √5) + 2⋅╲╱ ℯ + 4 "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F4.subs(x,1) - F4.subs(x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2.38910363799198$$"
      ],
      "text/plain": [
       "2.38910363799198"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(I4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int_{0}^{1} \\sqrt{e^{x} + 4}\\, dx$$"
      ],
      "text/plain": [
       "1               \n",
       "⌠               \n",
       "⎮    ________   \n",
       "⎮   ╱  x        \n",
       "⎮ ╲╱  ℯ  + 4  dx\n",
       "⌡               \n",
       "0               "
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2.38910363799198$$"
      ],
      "text/plain": [
       "2.38910363799198"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(I4.doit()) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.86 ms ± 195 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit  \n",
    "N(I4)\n",
    "# hier wird numerisch integriert"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "151 ms ± 1.72 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "N(I4.doit())\n",
    "# hier wird symbolisch integriert und dann numerisch ausgewertet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die unterliegende Numerik stammt bei `sympy` aus `mpmath`, welches beliebige Präzision erreichen kann.  In der CompLA werden dagegen `numpy` und `scipy` verwandt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2.38910363799198148358497100730434270931863307556131115296467$$"
      ],
      "text/plain": [
       "2.38910363799198148358497100730434270931863307556131115296467"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(I4, 60)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2.38910363799198148358497100730434270931863307556131115296467$$"
      ],
      "text/plain": [
       "2.38910363799198148358497100730434270931863307556131115296467"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(I4.doit(), 60)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "f5 = exp(-x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int e^{- x^{2}}\\, dx$$"
      ],
      "text/plain": [
       "⌠        \n",
       "⎮    2   \n",
       "⎮  -x    \n",
       "⎮ ℯ    dx\n",
       "⌡        "
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I5 = Integral(f5, x)\n",
    "I5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\pi} \\operatorname{erf}{\\left (x \\right )}}{2}$$"
      ],
      "text/plain": [
       "√π⋅erf(x)\n",
       "─────────\n",
       "    2    "
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I5.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`erf` ist Fehlerfunktion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "xn = np.linspace(-3, 3)\n",
    "fn = lambdify(x, erf(x), 'numpy')\n",
    "plt.plot(xn, fn(xn));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{\\left(- x^{2} + 1\\right)^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       "      1      \n",
       "─────────────\n",
       "          3/2\n",
       "⎛   2    ⎞   \n",
       "⎝- x  + 1⎠   "
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f7 = (1 - x**2)**Rational(-3,2)\n",
    "f7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int \\frac{1}{\\left(- x^{2} + 1\\right)^{\\frac{3}{2}}}\\, dx$$"
      ],
      "text/plain": [
       "⌠                 \n",
       "⎮       1         \n",
       "⎮ ───────────── dx\n",
       "⎮           3/2   \n",
       "⎮ ⎛   2    ⎞      \n",
       "⎮ ⎝- x  + 1⎠      \n",
       "⌡                 "
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I7 = Integral(f7, x)\n",
    "I7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} - \\frac{i x}{\\sqrt{x^{2} - 1}} & \\text{for}\\: \\left|{x^{2}}\\right| > 1 \\\\\\frac{x}{\\sqrt{- x^{2} + 1}} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧    -ⅈ⋅x           │ 2│    \n",
       "⎪ ───────────   for │x │ > 1\n",
       "⎪    ________               \n",
       "⎪   ╱  2                    \n",
       "⎪ ╲╱  x  - 1                \n",
       "⎨                           \n",
       "⎪      x                    \n",
       "⎪─────────────   otherwise  \n",
       "⎪   __________              \n",
       "⎪  ╱    2                   \n",
       "⎩╲╱  - x  + 1               "
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7 = I7.doit(conds='none')\n",
    "F7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left ( - \\frac{i x}{\\sqrt{x^{2} - 1}}, \\quad \\left|{x^{2}}\\right| > 1\\right ), \\quad \\left ( \\frac{x}{\\sqrt{- x^{2} + 1}}, \\quad \\mathrm{True}\\right )\\right )$$"
      ],
      "text/plain": [
       "⎛⎛   -ⅈ⋅x      │ 2│    ⎞  ⎛      x            ⎞⎞\n",
       "⎜⎜───────────, │x │ > 1⎟, ⎜─────────────, True⎟⎟\n",
       "⎜⎜   ________          ⎟  ⎜   __________      ⎟⎟\n",
       "⎜⎜  ╱  2               ⎟  ⎜  ╱    2           ⎟⎟\n",
       "⎝⎝╲╱  x  - 1           ⎠  ⎝╲╱  - x  + 1       ⎠⎠"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x}{\\sqrt{- x^{2} + 1}}$$"
      ],
      "text/plain": [
       "      x      \n",
       "─────────────\n",
       "   __________\n",
       "  ╱    2     \n",
       "╲╱  - x  + 1 "
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7a = F7.args[1][0]\n",
    "F7a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{\\left(- x^{2} + 1\\right)^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       "      1      \n",
       "─────────────\n",
       "          3/2\n",
       "⎛   2    ⎞   \n",
       "⎝- x  + 1⎠   "
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7a.diff(x).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{2 \\sqrt{3}}{3}$$"
      ],
      "text/plain": [
       "2⋅√3\n",
       "────\n",
       " 3  "
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I7a = Integral(f7, (x, Rational(-1,2), Rational(1,2)))\n",
    "I7a.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{2 \\sqrt{3}}{3}$$"
      ],
      "text/plain": [
       "2⋅√3\n",
       "────\n",
       " 3  "
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F7a.subs(x,Rational(1,2)) - F7a.subs(x, Rational(-1,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "2016 war die Stammfunktion noch falsch."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "f8 = 1 / (2 - cos(2*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\int \\frac{1}{- \\cos{\\left (2 x \\right )} + 2}\\, dx$$"
      ],
      "text/plain": [
       "⌠                 \n",
       "⎮       1         \n",
       "⎮ ───────────── dx\n",
       "⎮ -cos(2⋅x) + 2   \n",
       "⌡                 "
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I8 = Integral(f8, x)\n",
    "I8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{3} \\left(\\operatorname{atan}{\\left (\\sqrt{3} \\tan{\\left (x \\right )} \\right )} + \\pi \\lfloor{\\frac{x - \\frac{\\pi}{2}}{\\pi}}\\rfloor\\right)}{3}$$"
      ],
      "text/plain": [
       "   ⎛                    ⎢    π⎥⎞\n",
       "   ⎜                    ⎢x - ─⎥⎟\n",
       "   ⎜                    ⎢    2⎥⎟\n",
       "√3⋅⎜atan(√3⋅tan(x)) + π⋅⎢─────⎥⎟\n",
       "   ⎝                    ⎣  π  ⎦⎠\n",
       "────────────────────────────────\n",
       "               3                "
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F8 = I8.doit()\n",
    "F8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{2 \\sqrt{3} \\pi}{3}$$"
      ],
      "text/plain": [
       "2⋅√3⋅π\n",
       "──────\n",
       "  3   "
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I8a = Integral(f8, (x, 0, 2*pi))\n",
    "I8a.doit() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "F8n = lambdify(x, F8, 'numpy')\n",
    "xn = np.linspace(0, 2*np.pi)\n",
    "plt.plot(xn, F8n(xn));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{2 \\sqrt{3} \\pi}{3}$$"
      ],
      "text/plain": [
       "2⋅√3⋅π\n",
       "──────\n",
       "  3   "
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F8.subs(x, 2*pi) - F8.subs(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sqrt(3)*(atan(sqrt(3)*tan(x)) + pi*floor((x - pi/2)/pi))/3\n"
     ]
    }
   ],
   "source": [
    "print(F8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def null(x):\n",
    "    return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{3} \\operatorname{atan}{\\left (\\sqrt{3} \\tan{\\left (x \\right )} \\right )}}{3}$$"
      ],
      "text/plain": [
       "√3⋅atan(√3⋅tan(x))\n",
       "──────────────────\n",
       "        3         "
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = F8.replace(floor, null)  # replace ist mächtiger als subs\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [],
   "source": [
    "F8n2 = lambdify(x, a, 'numpy')\n",
    "plt.plot(xn, F8n2(xn));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Der `floor`-Term hebt die Sprünge auf"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}