examples/Giac_examples.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Giac.jl examples\n",
    "\n",
    "Organized according to the [HP Prime user guide](http://support.hp.com/us-en/product/HP-Prime-Graphing-Calculator/5367459/model/5367460/manuals/).\n",
    "\n",
    "## Algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "using Giac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x,y,z,n,k,a,b,c,d)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@giac x y z n k a b c d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### simplify\n",
    "\n",
    "Simplifies an expression.\n",
    "```\n",
    "simplify(Expr)\n",
    "```\n",
    "Note: rational numbers must be exlicitly transferred to Giac. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/4*pi"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(4*atan(giac(1//5))-atan(giac(1//239)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*cos(x)^2-2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(texpand((sin(3*x)+sin(7*x))/sin(5*x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*cos(x)^2-2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(texpand((cos(3*x)+cos(7*x))/cos(5*x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### collect\n",
    "Integer factorization of a polynomial (or of a list of poly).\n",
    "```\n",
    "collect(Poly or LstPoly)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*(x-1)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(x+2*x+1-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x-2)^2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(x^2-9*x+5*x+3+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x^2-2,(x-2)*(x+2)]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect([x^2-2,x^2-4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### expand\n",
    "Full distribution of * and / over + and -\n",
    "```\n",
    "expand(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x*z+y*z+x+y"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x+y)*(z+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "a/d+b/d+c/d"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((a+b+c)/d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x*y^2+x*z^2+y*z^2+x^2*y+x^2*z+y^2*z+2*x*y*z"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((y+x)*(z+y)*(x+z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^4+12*x^3+54*x^2+108*x+81"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x+3)^4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*x^3-15*x^2+22*x-13"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((2*x-2*1)*(x^2-3*x+2)+(x^2-2*x+3)*(2*x-3*1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### factor\n",
    "Factorizes a polynomial.\n",
    "```\n",
    "factor(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x-1)*(x+1)*(x^2+1)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x^4-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x-sqrt(2))*(x+sqrt(2))*(x^2+2)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x^4-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x-sqrt(2))*(x+sqrt(2))*(x^2+2)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x^4-4,sqrt(giac(2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x+3)^4"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x^4+12*x^3+54*x^2+108*x+81)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### subst\n",
    "Substitutes a value for a variable in an expression.\n",
    "```\n",
    "subst(Expr,Var(v)=>value(a))\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3/5"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "subst(x/(4-x^2),x=>3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning, replacing 2 by 3, a substitution variable should perhaps be purged.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1/(y^2+y^2)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "subst(1/(x^2+y^2),x=>2,y=>3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Error: Too many arguments: subst\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "undef"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "subst(1/(x^2+y^2+z^2),x=>2,y=>3,z=>1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### partfrac\n",
    "Performs partial fraction decomposition on a fraction.\n",
    "```\n",
    "partfrac(RatFrac)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/2/(x-2)-1/2/(x+2)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partfrac(x/(4-x^2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1+3/(x-2)-2/(x-1)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partfrac((x^2-2*x+3)/(x^2-3*x+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-a/b/z+a/b/(z-b)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partfrac(a/(z*(z-b)),z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algebra-Extract"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### numer\n",
    "Returns the numerator of the simplified fraction.\n",
    "```\n",
    "numer(Frac(a/b) or RatFrac)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numer(giac(25//15))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2+x+1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numer((x^3-1)/(x^2-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^3+x^2-1"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numer(1+(x^3-1)/x^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### denom\n",
    "Returns the denominator of the simplified fraction.\n",
    "```\n",
    "denom(Frac(a/b) or RatFrac)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denom(giac(25//15))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x+1"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denom((x^3-1)/(x^2-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denom(1+(x^3-1)/x^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### equal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2-1=(2*x+3)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "equal(x^2-1, 2*x+3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate equation using ⩦ sign (write \\eqdot then press TAB key):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2-1=(2*x+3)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x^2-1 ⩦ 2*x+3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### left\n",
    "Returns the left part of an equation.\n",
    "```\n",
    "left(Equal(a⩦b))\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2-1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "left(x^2-1 ⩦ 2*x+3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### right\n",
    "Returns the right part of an equation.\n",
    "```\n",
    "right(Equal(a⩦b))\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*x+3"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "right(x^2-1 ⩦ 2*x+3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### diff\n",
    "Returns the derivative with respect to the 2nd argument.\n",
    "```\n",
    "diff(Expr or Fnc,[SeqVar or LstVar])\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*x^2-1"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x^3-x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*x^2-1"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x^3-x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6*x"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x^3-x,x,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2*y^3*exp(x*y)+6*x*y^2*exp(x*y)+6*y*exp(x*y)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(exp(x*y),x,x,x,y,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x*y+z*y,y,z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x+z,y]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x*y+z*y,[y,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       " (x)->sin(2*x)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@giac f(x)=sin(2x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*cos(2*x)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f'(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "diff(y(x),x)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y'(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### integrate\n",
    "Definite or indefinite integral. \n",
    "```\n",
    "integrate(Expr,[Var(x)],[Real(a)],[Real(b)])\n",
    "``` \n",
    "Note that this function cannot be named 'int', because Julia doesn't allow to overload the int function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ln(abs(x))"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(1/x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/2*atan(z/2)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(1/(4+z^2),z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/4*(2*atan(2)+ln(3))+1/4*(2*atan(3)-ln(2)+ln(4))"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(1/(1-x^4),x,2,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### limit\n",
    "Limit of an expression at a point (or plus_inf or minus_inf, d=-1 for left lim and +1 for right lim).\n",
    "```\n",
    "limit(Expr,Var=>Val,[Dir(d)])\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((n*tan(x)-tan(n*x))/(sin(n*x)-n*sin(x)),x=>0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/3"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/(x^2-3*x),x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "+infinity"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((2*x-1)/exp(1/(x-1)),x,plus_inf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sign(x),x=>0,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sign(x),x=>0,-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### series\n",
    "Series expansion at finite or infinite points.\n",
    "```\n",
    "series(Expr,Var=>Val,[Order])\n",
    "series(Expr,Vars,Vals,[Order])\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2+x-2*x^2-x^3+3*x^4+x^5+x^6*order_size(x)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series((x^4+x+2)/(x^2+1),x=>0, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-x-y+x^2+2*x*y+y^2-x^3-3*x^2*y-3*x*y^2-y^3+x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4-x^5-5*x^4*y-10*x^3*y^2-10*x^2*y^3-5*x*y^4-y^5"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(1/(1+x+y),[x,y],[0,0],5) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### sum\n",
    "Discrete sum (with 2 or 4 arguments return then sum from a to b if a<=b or of the opposite of the\n",
    "sum from b+1 to a-1 if a>b+1 or 0 if a=b+1) or the discrete primitive or sum of the elements of a\n",
    "list or a sequence.\n",
    "```\n",
    "sum(Expr,Var,VarMin(a),VarMax(b),[VarStep(p)])\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "238357395880861/150117385017600"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(1/n^2,n,1,17)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "141433003757/117279207045"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(1/n^2,n,1,17,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/6*pi^2"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(1/n^2,n,1,infinity)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([1,2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-cos(n*x)*cos(x)+cos(n*x)-sin(n*x)*sin(x))/(2*cos(x)-2)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(cos(n*x),n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*n^3-3*n^2+n)/6"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(n*n,n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus-Differental"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### curl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[z-x,0,z-2*x]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "curl([2*x*y,x*z,y*z],[x,y,z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### divergence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*x+3*z^2+1"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "divergence([x^2+y,x+z+y,z^3+x^2],[x,y,z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2*2*x*y-z^3,2*x^2,-x*3*z^2]"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grad(2*x^2*y-x*z^3,[x,y,z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### hessian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[4*y,4*x,-1],[2*2*x,0,0],[-1,0,0]]"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hessian(2*x^2*y-x*z,[x,y,z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus-Integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### preval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "preval(x^2-2,2,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "preval(integrate(y,y),0,1,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus-Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### sum_riemann"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ln(2)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_riemann(1/(n+k),[n,k])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi/4"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_riemann(n/(n^2+k^2),[n,k])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### taylor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-1/6*x^2+1/120*x^4+x^6*order_size(x)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(sin(x)/x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-1/6*x^2+x^4*order_size(x)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(sin(x)/x,x,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-1/6*x^2+1/120*x^4+x^6*order_size(x)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(sin(x)/x,x=>0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-1/6*x^2+1/120*x^4-1/5040*x^6+x^8*order_size(x)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(sin(x)/x,x=>0, 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### divpc\n",
    "n-degree Taylor-poly for the quotient of 2 polynomials.\n",
    "\n",
    "divpc(Poly,Poly,Intg(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "LoadError: UndefVarError: divpc not defined\nwhile loading In[71], in expression starting on line 1",
     "output_type": "error",
     "traceback": [
      "LoadError: UndefVarError: divpc not defined\nwhile loading In[71], in expression starting on line 1",
      ""
     ]
    }
   ],
   "source": [
    "divpc(x^4+x+2,x^2+1,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[-2,2]"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x^2-3⩦1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[-2]"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x^2-3⩦1, x+2⩦0],x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[[1/3,1/3,1/3]]"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([y-z⩦0,z-x⩦0,x-y⩦0,x-1+y+z⩦0],[x,y,z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### zeros"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2,-2]"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros(x^2-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2,-2]"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros(x^2-4,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1,1],[1,-1],[-1,1],[-1,-1]]"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros([x^2-1,x^2-y^2],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### cSolve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[-1,1,i,-i]"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cSolve(x^4-1⩦0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[-1,1,i,-i]"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cSolve(x^4-1⩦0, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list[[0,0],[2,-2]]"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cSolve([x^4-y^4,x+y⩦0,x^2⩦2*x],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### cZeros"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1,i,1,-i]"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cZeros(x^4-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1,1],[1,-1],[-1,1],[-1,-1]]"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cZeros([x^2-1,x^2-y^2],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### fSolve\n",
    "Numerical solution of an equation or a system of equation.\n",
    "```\n",
    "fSolve(Expr,Var,[Guess or Interval],[Method])\n",
    "```\n",
    "(Intervall and Method not yet implemented...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.739085133215"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fSolve(cos(x)⩦x,x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.739085133215"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fSolve(cos(x)⩦x,x=>0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Solving by bisection with change of variable x=tan(t) and t=-1.57..1.57. Try fsolve(equation,x=guess) for iterative solver or fsolve(equation,x=xmin..xmax) for bisection.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[0.739085133215]"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fSolve(cos(x)⩦x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Solving by bisection with change of variable x=tan(t) and t=-1.57..1.57. Try fsolve(equation,x=guess) for iterative solver or fsolve(equation,x=xmin..xmax) for bisection.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[0.739085133215]"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fSolve(cos(x)⩦x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.61803398875,-0.61803398875]"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fSolve([x^2+y-2,x+y^2-2],[x,y],[.5,-.5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### deSolve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solves a differential equation or a differential linear system with constant coefficients.\n",
    "```\n",
    "desolve(Eq,[TimeVar],FncVar)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "c_0*cos(x)+c_1*sin(x)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol=deSolve(diff(y(x),x,x)+y(x)⩦0, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17*cos(x)+c_1*sin(x)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "subst(sol, giac(\"c_0\")=>17)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "c_0*cos(x)+c_1*sin(x)"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deSolve(y''(x)+y(x)⩦0, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### linsolve\n",
    "Linear equations system solver.\n",
    "```\n",
    "linsolve(LstLinEq,LstVar)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3/2,-1/2,0]"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([x+y+z⩦1,x-y⩦2,2*x-z⩦3],[x,y,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(a*n-b)/(n^2-1),(-a+b*n)/(n^2-1)]"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([n*x+y⩦a,x+n*y⩦b],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### logcollect\n",
    "Collect logarithms. Applies $\\log(a)+n\\log(b)\\to\\log(ab^n)$. \n",
    "```\n",
    "logcollect(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ln(x*y^2)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logcollect(log(x)+2*log(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### powexpand\n",
    "Applies $a^{b+c}\\to a^b a^c$.\n",
    "```\n",
    "powexpand(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2^x*2^y"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "powexpand(2^(x+y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### texpand\n",
    "Expand transcendental expressions.\n",
    "```\n",
    "texpand(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*cos(x)*sin(x)+exp(x)*exp(y)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "texpand(sin(2*x)+exp(x+y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite - exp & log"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### exp2pow\n",
    "Applies $\\mathrm{e}^{n\\log(x)}\\to x^n$.\n",
    "```\n",
    "exp2pow(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^3"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp2pow(exp(3*log(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pow2exp\n",
    "Converts powers to exponentials.\n",
    "```\n",
    "pow2exp(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(b*ln(a))"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pow2exp(a^b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### exp2trig\n",
    "Rewrites complex exponentials by sines and cosines.\n",
    "```\n",
    "exp2trig(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)+i*sin(x)"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp2trig(exp(1im*x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### expexpand\n",
    "Expands exponentials.\n",
    "```\n",
    "expexpand(Expr)\n",
    "```\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(x)^3*exp(y)^2"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expexpand(exp(3*x+2*y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite - Sine"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### asin2acos\n",
    "Replaces $\\mathrm{asin}(x)$ by $\\pi/2-\\mathrm{acos}(x)$.\n",
    "```\n",
    "asin2acos(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi/2-acos(x)+acos(x)"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "asin2acos(acos(x)+asin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### asin2atan\n",
    "Replaces $\\mathrm{asin}(x)$ by $\\mathrm{atan}(x/\\sqrt{1-x^2})$.\n",
    "```\n",
    "asin2atan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*atan(x/sqrt(1-x^2))"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "asin2atan(2*asin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### sin2costan\n",
    "Replaces $\\sin(x)$ by $\\cos(x)\\tan(x)$ \n",
    "```\n",
    "sin2costan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tan(x)*cos(x)"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin2costan(sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite - Cosine"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### acos2asin\n",
    "Replaces $\\mathrm{acos}(x)$ by $\\pi/2-\\mathrm{asin}(x)$.\n",
    "```\n",
    "acos2asin(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi/2-asin(x)+asin(x)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acos2asin(acos(x)+asin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### acos2atan\n",
    "Replaces $\\mathrm{acos}(x)$ by $\\pi/2-\\mathrm{atan}(x/\\sqrt{1-x^2})$.\n",
    "```\n",
    "acos2atan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*(pi/2-atan(x/sqrt(1-x^2)))"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acos2atan(2*acos(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### cos2sintan\n",
    "Replaces $\\cos(x)$ by $\\sin(x)/\\tan(x)$ \n",
    "```\n",
    "cos2sintan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(x)/tan(x)"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos2sintan(cos(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite - Tangent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### atan2asin\n",
    "Replaces $\\mathrm{atan}(x)$ by $\\mathrm{acos}(x/\\sqrt{1+x^2})$.\n",
    "```\n",
    "atan2asin(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "asin(2*x/sqrt(1+(2*x)^2))"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "atan2asin(atan(2*x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### atan2acos\n",
    "Replaces $\\mathrm{atan}(x)$ by $\\pi/2$-$\\mathrm{acos}(x/\\sqrt{1+x^2})$.\n",
    "```\n",
    "atan2acos(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi/2-acos(2*x/sqrt(1+(2*x)^2))"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "atan2acos(atan(2*x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### tan2sincos\n",
    "Replaces $\\tan(x)$ by $\\sin(x)/\\cos(x)$.\n",
    "```\n",
    "tan2sincos(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(x)/cos(x)"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tan2sincos(tan(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### halftan\n",
    "Rewrites $\\sin(x)$, $\\cos(x)$, and $\\tan(x)$ as function of $\\tan(x/2)$.\n",
    "```\n",
    "halftan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((1-tan(x/2)^2)/(tan(x/2)^2+1),2*tan(x/2)/(tan(x/2)^2+1),2*tan(x/2)/(1-tan(x/2)^2))"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "halftan(cos(x)), halftan(sin(x)), halftan(tan(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rewrite - Trig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trigsin\n",
    "Simplifies expression  using $\\sin(x)^2+\\cos(x)^2=1$ and $\\tan(x)=\\sin(x)/\\cos(x)$. $\\sin(x)$ is given precedence over $\\cos(x)$ and $\\tan(x)$ in the result.\n",
    "```\n",
    "trigsin(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(x)^4-sin(x)^2+1"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsin(cos(x)^4+sin(x)^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trigcos\n",
    "Simplifies expression  using $\\sin(x)^2+\\cos(x)^2=1$ and $\\tan(x)=\\sin(x)/\\cos(x)$. \n",
    "$\\cos(x)$ is given precedence over $\\sin(x)$ and $\\tan(x)$ in the result.\n",
    "```\n",
    "trigcos(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)^4-cos(x)^2+1"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigcos(cos(x)^4+sin(x)^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trigtan\n",
    "Simplifies expression  using $\\sin(x)^2+\\cos(x)^2=1$ and $\\tan(x)=\\sin(x)/\\cos(x)$. \n",
    "$\\tan(x)$ is given precedence over $\\sin(x)$ and $\\cos(x)$ in the result.\n",
    "```\n",
    "trigtan(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tan(x)^4+tan(x)^2+1)/(tan(x)^4+2*tan(x)^2+1)"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigtan(cos(x)^4+sin(x)^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### atrig2log\n",
    "Rewrites inverse trigonometric functions as logarithms.\n",
    "```\n",
    "atrig2log(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "i*ln((i+x)/(i-x))/2"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "atrig2log(atan(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### tlin\n",
    "Trigonometric linearization.\n",
    "```\n",
    "tlin(ExprTrig)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3/4*sin(x)-1/4*sin(3*x)"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tlin(sin(x)^3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### tcollect\n",
    "Collect trigonometric functions with like arguments.\n",
    "```\n",
    "tCollect(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(2)*cos(x-1/4*pi)"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tcollect(sin(x)+cos(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trigexpand\n",
    "Expands trigonometric expressions.\n",
    "```\n",
    "trigexpand(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4*cos(x)^2-1)*sin(x)"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigexpand(sin(3*x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trig2exp\n",
    "Rewrites trigonometric functions by complex exponentials.\n",
    "```\n",
    "trig2exp(Expr)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(exp(i*x)-1/exp(i*x))/(2*i)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trig2exp(sin(x))"
   ]
  }
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