SymPy_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "removed-monte",
   "metadata": {},
   "source": [
    "# SymPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "collectible-somerset",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "finished-table",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "x**2 + 2*x + 1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z, t = symbols('x y z t')\n",
    "f, g, h = symbols('f g h', cls=Function)\n",
    "\n",
    "simplify(x**2 + 2*x + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "differential-destination",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "x**2 + 2*x + 1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + 1)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "compressed-transport",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} - x - 6$"
      ],
      "text/plain": [
       "x**2 - x - 6"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + 2)*(x - 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "internal-sunday",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + 1)*(x - 2) - (x - 1)*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ancient-attempt",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x - 1\\right) \\left(x^{2} + 1\\right)$"
      ],
      "text/plain": [
       "(x - 1)*(x**2 + 1)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**3 - x**2 + x - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "devoted-sympathy",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle z \\left(x + 2 y\\right)^{2}$"
      ],
      "text/plain": [
       "z*(x + 2*y)**2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**2*z + 4*x*y*z + 4*y**2*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "solved-eligibility",
   "metadata": {},
   "source": [
    "For polynomials, factor() is the opposite of expand(). factor() uses a complete multivariate factorization algorithm over the rational numbers, which means that each of the factors returned by factor() is guaranteed to be irreducible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "suspected-hotel",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, [(z, 1), (x + 2*y, 2)])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_list(x**2*z + 4*x*y*z + 4*y**2*z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "magnetic-boulder",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right)^{2}$"
      ],
      "text/plain": [
       "(sin(x) + cos(x))**2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(cos(x)**2 + 2*cos(x)*sin(x) + sin(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bound-caribbean",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2} z + 2 x^{2} + x y + x - 3$"
      ],
      "text/plain": [
       "x**3 - x**2*z + 2*x**2 + x*y + x - 3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "seeing-wonder",
   "metadata": {},
   "source": [
    "cancel() will take any rational function and put it into the standard canonical form, 𝑝𝑞, where 𝑝 and 𝑞 are expanded polynomials with no common factors, and the leading coefficients of 𝑝 and 𝑞 do not have denominators (i.e., are integers)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "worthy-procedure",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x + 1}{x}$"
      ],
      "text/plain": [
       "(x + 1)/x"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " cancel((x**2 + 2*x + 1)/(x**2 + x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "greenhouse-import",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\frac{3 x}{2} - 2}{x - 4} + \\frac{1}{x}$"
      ],
      "text/plain": [
       "(3*x/2 - 2)/(x - 4) + 1/x"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = 1/x + (3*x/2 - 2)/(x - 4)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "compound-variance",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3 x^{2} - 2 x - 8}{2 x^{2} - 8 x}$"
      ],
      "text/plain": [
       "(3*x**2 - 2*x - 8)/(2*x**2 - 8*x)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cancel(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "agricultural-monster",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(x - 2\\right) \\left(3 x + 4\\right)}{2 x \\left(x - 4\\right)}$"
      ],
      "text/plain": [
       "(x - 2)*(3*x + 4)/(2*x*(x - 4))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "crazy-pocket",
   "metadata": {},
   "source": [
    "apart() performs a partial fraction decomposition on a rational function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "solar-thought",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{4 x^{3} + 21 x^{2} + 10 x + 12}{x^{4} + 5 x^{3} + 5 x^{2} + 4 x}$"
      ],
      "text/plain": [
       "(4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "artificial-following",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x - 1}{x^{2} + x + 1} - \\frac{1}{x + 4} + \\frac{3}{x}$"
      ],
      "text/plain": [
       "(2*x - 1)/(x**2 + x + 1) - 1/(x + 4) + 3/x"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apart(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "progressive-sitting",
   "metadata": {},
   "source": [
    "# Trigonometric Simplification\n",
    "- Note SymPy follows Python’s naming conventions for inverse trigonometric functions, which is to append an a to the front of the function’s name. For example, the inverse cosine, or arc cosine, is called acos()."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "continued-excuse",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{acos}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "acos(x)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "caroline-antenna",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(acos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "postal-positive",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\pi}{2}$"
      ],
      "text/plain": [
       "pi/2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "asin(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "historical-dialogue",
   "metadata": {},
   "source": [
    "## trigsimp\n",
    "To simplify expressions using trigonometric identities, use trigsimp()."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "pressing-updating",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(sin(x)**2 + cos(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "growing-solomon",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\cos{\\left(4 x \\right)}}{2} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "cos(4*x)/2 + 1/2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "fancy-bones",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cosh{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "cosh(2*x)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# hyperbolic trig func\n",
    "trigsimp(cosh(x)**2 + sinh(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "coupled-touch",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cosh{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cosh(x)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(sinh(x)/tanh(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "dramatic-reputation",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)} \\cos{\\left(y \\right)} + \\sin{\\left(y \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)*cos(y) + sin(y)*cos(x)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_trig(sin(x + y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "packed-stations",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\tan{\\left(x \\right)}}{1 - \\tan^{2}{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "2*tan(x)/(1 - tan(x)**2)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_trig(tan(2*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "heard-march",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x + y \\right)}$"
      ],
      "text/plain": [
       "sin(x + y)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(sin(x)*cos(y) + sin(y)*cos(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "false-local",
   "metadata": {},
   "source": [
    "## Powers\n",
    "\n",
    "Before we introduce the power simplification functions, a mathematical discussion on the identities held by powers is in order. There are three kinds of identities satisfied by exponents\n",
    "\n",
    "𝑥𝑎𝑥𝑏=𝑥𝑎+𝑏\n",
    "\n",
    "𝑥𝑎𝑦𝑎=(𝑥𝑦)𝑎\n",
    "\n",
    "(𝑥𝑎)𝑏=𝑥𝑎𝑏"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "identical-moldova",
   "metadata": {},
   "source": [
    "|Identity| Sufficient conditions to hold|Counterexample when conditions are not met|Important consequences|\n",
    "|--------|------------------------------|---------------|------------------|\n",
    "|𝑥𝑎𝑥𝑏=𝑥𝑎+𝑏| Always true | None | None|\n",
    "|𝑥𝑎𝑦𝑎=(𝑥𝑦)𝑎 | 𝑥,𝑦≥0 and 𝑎∈ℝ | (−1)1/2(−1)1/2≠(−1⋅−1)1/2|𝑥‾‾√𝑦√≠𝑥𝑦‾‾‾√ in general|\n",
    "|(𝑥𝑎)𝑏=𝑥𝑎𝑏 | 𝑏∈ℤ | ((−1)2)1/2≠(−1)2⋅1/2 | 𝑥2‾‾‾√≠𝑥 and 1𝑥‾‾√≠1𝑥√ in general|\n",
    "\n",
    "This is important to remember, because by default, SymPy will not perform simplifications if they are not true in general."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "secondary-surname",
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = symbols('x y', positive=True)\n",
    "a, b = symbols('a b', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "beneficial-humanitarian",
   "metadata": {},
   "outputs": [],
   "source": [
    "z, t, c = symbols('z t c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "bronze-biography",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{a + b}$"
      ],
      "text/plain": [
       "x**(a + b)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "powsimp(x**a*x**b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "purple-chemistry",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x y\\right)^{a}$"
      ],
      "text/plain": [
       "(x*y)**a"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "powsimp(x**a*y**a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "small-balance",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{x} \\sqrt{y}$"
      ],
      "text/plain": [
       "sqrt(x)*sqrt(y)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "economic-quantum",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{a} x^{b}$"
      ],
      "text/plain": [
       "x**a*x**b"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_power_exp(x**(a + b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "least-execution",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{5}$"
      ],
      "text/plain": [
       "x**5"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x**2*x**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "veterinary-advertising",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{a b}$"
      ],
      "text/plain": [
       "x**(a*b)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "powdenest((x**a)**b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "conservative-rabbit",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(z^{a}\\right)^{b}$"
      ],
      "text/plain": [
       "(z**a)**b"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "powdenest((z**a)**b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "completed-infrared",
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = symbols('x y', positive=True)\n",
    "n = symbols('n', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "mexican-removal",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(x \\right)} + \\log{\\left(y \\right)}$"
      ],
      "text/plain": [
       "log(x) + log(y)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_log(log(x*y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "positive-indicator",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(x \\right)} - \\log{\\left(y \\right)}$"
      ],
      "text/plain": [
       "log(x) - log(y)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_log(log(x/y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "registered-teens",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle n \\log{\\left(x \\right)}$"
      ],
      "text/plain": [
       "n*log(x)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_log(log(x**2))\n",
    "expand_log(log(x**n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "gorgeous-saudi",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle n!$"
      ],
      "text/plain": [
       "factorial(n)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z = symbols('x y z')\n",
    "k, m, n = symbols('k m n')\n",
    "\n",
    "factorial(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "congressional-eating",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle {\\binom{n}{k}}$"
      ],
      "text/plain": [
       "binomial(n, k)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binomial(n, k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "rural-burke",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\Gamma\\left(z\\right)$"
      ],
      "text/plain": [
       "gamma(z)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gamma(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "imported-algebra",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + \\frac{1}{z}$"
      ],
      "text/plain": [
       "x + 1/z"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "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\n",
    "\n",
    "list_to_frac([x, y, z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "alone-duplicate",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5}{4}$"
      ],
      "text/plain": [
       "5/4"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_to_frac([1, 2, 3, 4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "young-allen",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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