{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note**: This is taken from the [Maxima Tutorial](http://maxima.sourceforge.net/docs/tutorial/en/gaertner-tutorial-revision/Contents.htm), specifically [this page](http://maxima.sourceforge.net/docs/tutorial/en/gaertner-tutorial-revision/Pages/Intro0001.htm). Same licensing applies.\n",
    "\n",
    "# A First Look at Maxima\n",
    "\n",
    "Maxima can be used as a powerful calucator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{2}$}2439\\]"
      ],
      "text/plain": [
       "(%o2)                                2439"
      ],
      "text/x-maxima": [
       "2439"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "144*17 - 9;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maxima can compute with very large numbers. The following expample computes the 25th power of 144:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}910043815000214977332758527534256632492715260325658624\\]"
      ],
      "text/plain": [
       "(%o3)       910043815000214977332758527534256632492715260325658624"
      ],
      "text/x-maxima": [
       "910043815000214977332758527534256632492715260325658624"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "144^25;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is more than a pocket calculator can do!\n",
    "\n",
    "Now we compute the 25th root of that result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}144\\]"
      ],
      "text/plain": [
       "(%o4)                                 144"
      ],
      "text/x-maxima": [
       "144"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " %^(1/25);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The percent sign is a special variable, its value is always the last result. The arrow is the exponentiation operator. In our input we used some elementary mathematical knowledge: we wrote the root as a power with a fractional exponent.\n",
    "\n",
    "But computer algebra is more than just computation with numbers. It is computation with symbols.\n",
    "\n",
    "Let us play with a polynomial in two variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}\\left(2\\,y+x\\right)^4\\]"
      ],
      "text/plain": [
       "                                           4\n",
       "(%o5)                             (2 y + x)"
      ],
      "text/x-maxima": [
       "(2*y+x)^4"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 2*y)^4;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}16\\,y^4+32\\,x\\,y^3+24\\,x^2\\,y^2+8\\,x^3\\,y+x^4\\]"
      ],
      "text/plain": [
       "                       4         3       2  2      3      4\n",
       "(%o6)              16 y  + 32 x y  + 24 x  y  + 8 x  y + x"
      ],
      "text/x-maxima": [
       "16*y^4+32*x*y^3+24*x^2*y^2+8*x^3*y+x^4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}\\left(2\\,y+x\\right)^4\\]"
      ],
      "text/plain": [
       "                                           4\n",
       "(%o7)                             (2 y + x)"
      ],
      "text/x-maxima": [
       "(2*y+x)^4"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(%);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maxima can compute derivatives:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}\\cos ^2x-\\sin ^2x\\]"
      ],
      "text/plain": [
       "                                  2         2\n",
       "(%o8)                          cos (x) - sin (x)"
      ],
      "text/x-maxima": [
       "cos(x)^2-sin(x)^2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x)*cos(x), x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}2\\,\\cos ^2x-1\\]"
      ],
      "text/plain": [
       "                                      2\n",
       "(%o9)                            2 cos (x) - 1"
      ],
      "text/x-maxima": [
       "2*cos(x)^2-1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}-4\\,\\cos x\\,\\sin x\\]"
      ],
      "text/plain": [
       "(%o10)                         - 4 cos(x) sin(x)"
      ],
      "text/x-maxima": [
       "-4*cos(x)*sin(x)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(%, x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}-4\\,\\cos x\\,\\sin x\\]"
      ],
      "text/plain": [
       "(%o11)                         - 4 cos(x) sin(x)"
      ],
      "text/x-maxima": [
       "-4*cos(x)*sin(x)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x)*cos(x), x, 2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maxima can rewrite trigonometric expressions in a canonical form, namely as finite Fourier sums:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{12}$}\\frac{\\sin \\left(5\\,x\\right)-5\\,\\sin \\left(3\\,x\\right)+10\\,\\sin x}{16}\\]"
      ],
      "text/plain": [
       "                       sin(5 x) - 5 sin(3 x) + 10 sin(x)\n",
       "(%o12)                 ---------------------------------\n",
       "                                      16"
      ],
      "text/x-maxima": [
       "(sin(5*x)-5*sin(3*x)+10*sin(x))/16"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigreduce (sin(x)^5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maxima can compute indefinite integrals:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{13}$}-\\frac{\\log \\left(x^2+2\\,x+4\\right)}{8}+\\frac{\\arctan \\left(\\frac{2\\,x+2}{2\\,\\sqrt{3}}\\right)}{4\\,\\sqrt{3}}+\\frac{\\log \\left(x-2\\right)}{4}\\]"
      ],
      "text/plain": [
       "                                           2 x + 2\n",
       "                     2               atan(---------)\n",
       "                log(x  + 2 x + 4)         2 sqrt(3)    log(x - 2)\n",
       "(%o13)       (- -----------------) + --------------- + ----------\n",
       "                        8               4 sqrt(3)          4"
      ],
      "text/x-maxima": [
       "(-log(x^2+2*x+4)/8)+atan((2*x+2)/(2*sqrt(3)))/(4*sqrt(3))+log(x-2)/4"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " integrate((x + 1)/(x^3 - 8), x);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a longer example that shows that Maxima can compute quite complicated integrals and can also often reconstruct the given integrand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{14}$}\\left[ m>4 \\right] \\]"
      ],
      "text/plain": [
       "(%o14)                              [m > 4]"
      ],
      "text/x-maxima": [
       "[m > 4]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "assume(m>4);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{15}$}\\frac{b^3\\,x^{m+4}}{m+4}+\\frac{3\\,a\\,b^2\\,x^{m+3}}{m+3}+\\frac{3\\,a^2\\,b\\,x^{m+2}}{m+2}+\\frac{a^3\\,x^{m+1}}{m+1}\\]"
      ],
      "text/plain": [
       "              3  m + 4        2  m + 3      2    m + 2    3  m + 1\n",
       "             b  x        3 a b  x        3 a  b x        a  x\n",
       "(%o15)       --------- + ------------- + ------------- + ---------\n",
       "               m + 4         m + 3           m + 2         m + 1"
      ],
      "text/x-maxima": [
       "(b^3*x^(m+4))/(m+4)+(3*a*b^2*x^(m+3))/(m+3)+(3*a^2*b*x^(m+2))/(m+2)\n",
       "                   +(a^3*x^(m+1))/(m+1)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x^m*(a + b*x)^3, x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{16}$}b^3\\,x^{m+3}+3\\,a\\,b^2\\,x^{m+2}+3\\,a^2\\,b\\,x^{m+1}+a^3\\,x^{m}\\]"
      ],
      "text/plain": [
       "                3  m + 3        2  m + 2      2    m + 1    3  m\n",
       "(%o16)         b  x      + 3 a b  x      + 3 a  b x      + a  x"
      ],
      "text/x-maxima": [
       "b^3*x^(m+3)+3*a*b^2*x^(m+2)+3*a^2*b*x^(m+1)+a^3*x^m"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(%, x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{17}$}x^{m}\\,\\left(b\\,x+a\\right)^3\\]"
      ],
      "text/plain": [
       "                                  m          3\n",
       "(%o17)                           x  (b x + a)"
      ],
      "text/x-maxima": [
       "x^m*(b*x+a)^3"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(%);"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Maxima",
   "language": "maxima",
   "name": "maxima"
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