SciPy_SymPy/tutorial_exercises/Simplification Solutions.ipynb

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    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Simplification Solutions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Boilerplate to make the doctester work."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sys\n",
      "import os\n",
      "sys.path.insert(1, os.path.join(os.path.pardir, \"ipython_doctester\"))\n",
      "from sympy import *\n",
      "from ipython_doctester import test\n",
      "x, y, z = symbols('x y z')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 68
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For each exercise, fill in the function according to its docstring. Execute the cell to see if you did it right. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Polynomial/Rational Function Simplification"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In each exercise, apply specific simplification functions to get the desired result."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def polysimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> polysimp1(cos(x)*sin(x) + cos(x))\n",
      "    (sin(x) + 1)*cos(x)\n",
      "    >>> polysimp1(cos(x)*sin(x) + cos(x) + 1)\n",
      "    (sin(x) + 1)*cos(x) + 1\n",
      "    \"\"\"\n",
      "    return collect(expr, cos(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def polysimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> polysimp2((2*x + 1)/(x**2 + x))\n",
      "    1/(x + 1) + 1/x\n",
      "    >>> polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))\n",
      "    1/(x**2 + 2*x + 1) + 1/x\n",
      "    \"\"\"\n",
      "    return expand(apart(expr))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Powers"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In each exercise, apply specific simplification functions to get the desired result. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def powersimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> powersimp1(exp(x)*(exp(y) + 1))\n",
      "    exp(x) + exp(x + y)\n",
      "    \"\"\"\n",
      "    return powsimp(expand(expr))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def powersimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> powersimp2(2**x*x**x)\n",
      "    (2*x)**x\n",
      "    >>> powersimp2(x**x*x**x)\n",
      "    (x**2)**x\n",
      "    \"\"\"\n",
      "    return powsimp(expr, force=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def powersimp3(expr):\n",
      "    \"\"\"\n",
      "    >>> a, b, c = symbols('a b c')\n",
      "    >>> powersimp3((a**b)**c)\n",
      "    a**(b*c)\n",
      "    >>> powersimp3((a**b)**(c + 1))\n",
      "    a**(b*c + b)\n",
      "    \"\"\"\n",
      "    return powdenest(expand_power_exp(expr), force=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Logs"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def logsimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> a, b = symbols('a b', positive=True)\n",
      "    >>> logsimp1(log(x**y*a**b))\n",
      "    y*log(x) + log(a**b)\n",
      "    >>> logsimp1(log(x*y*a*b))\n",
      "    log(x) + log(y) + log(a*b)\n",
      "    \"\"\"\n",
      "    return logcombine(expand_log(expr, force=True))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Miscellaneous  "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def miscsimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> miscsimp1(sin(x + y))\n",
      "    2*(-tan(x/2)**2 + 1)*tan(y/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1)) + 2*(-tan(y/2)**2 + 1)*tan(x/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1))\n",
      "    \"\"\"\n",
      "    return expand_trig(expr).rewrite(tan)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def miscsimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> miscsimp2(gamma(x + 4))\n",
      "    x**4*gamma(x) + 6*x**3*gamma(x) + 11*x**2*gamma(x) + 6*x*gamma(x)\n",
      "    \"\"\"\n",
      "    return expand(expand_func(expr))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
       ],
       "output_type": "display_data"
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Continued Fractions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def list_to_frac(l):\n",
      "    expr = Integer(0)\n",
      "    for i in reversed(l[1:]):\n",
      "        expr += i\n",
      "        expr = 1/expr\n",
      "    return l[0] + expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a0, a1, a2, a3, a4 = symbols('a0:5')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Determine the list used to create the continued fraction $$\\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}.$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@test\n",
      "def continued_frac():\n",
      "    \"\"\"\n",
      "    Determine the original list used to create the fraction.  \n",
      "\n",
      "    Return the original list from this function.\n",
      "\n",
      "    >>> orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)\n",
      "    >>> pprint(orig_frac, use_unicode=False, wrap_line=False)\n",
      "    a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3\n",
      "    -------------------------------------------------------------------------\n",
      "                     a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1                 \n",
      "    >>> cancel(list_to_frac(continued_frac())) == orig_frac\n",
      "    True\n",
      "    \"\"\"\n",
      "    return [a3, a4, a0, a2, a1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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       "html": [
        "\n",
        "      <p style=\"color:green;font-size:250%;font-weight=bold\">Success!</p>\n",
        "      "
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       "output_type": "display_data"
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
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