Equation Solving & Taylor Series.ipynb

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    "# Equation Solving\n",
    "\n",
    "First we will learn how to an equation in t when equated to 0\n",
    "$$v_0t - \\frac{1}{2}gt^2 = 0$$"
   ]
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   "execution_count": 1,
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   "source": [
    "from sympy import (\n",
    "    symbols, #Define mathematical symbols for symbolic math\n",
    "    solve, \n",
    "    Rational, #define rational numbers\n",
    "    lambdify #Turn symbolic expressions to Python functions\n",
    ")"
   ]
  },
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   "execution_count": 2,
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   "outputs": [],
   "source": [
    "t, v0, g = symbols('t v0 g')"
   ]
  },
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   "source": [
    "y = v0*t - Rational(1, 2)*g*t**2"
   ]
  },
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   "source": [
    "roots = solve(y , t)"
   ]
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 2*v0/g]"
      ]
     },
     "execution_count": 5,
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     "output_type": "execute_result"
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   "source": [
    "roots"
   ]
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   "source": [
    "We can easily check the answer by inserting the values in the expression. This is done as shown below"
   ]
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   "source": [
    "y.subs(t, roots[0])"
   ]
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       "0"
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   "source": [
    "y.subs(t, roots[1])"
   ]
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   "source": [
    "# Taylor Polynomial\n",
    "\n",
    "A taylor polynomial of order `n` for an expression `e` in a variable `t` around the point `t0` is computed by `e.series(t, t0, n)` "
   ]
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   "source": [
    "Let's test this on $e^t$ and $e^{sin(t)}$"
   ]
  },
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   "execution_count": 9,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + t + t**2/2 + O(t**3)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import sin, cos, exp\n",
    "f = exp(t)\n",
    "f.series(t, 0, 3)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 10,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + t + t**2/2 - t**4/8 - t**5/15 - t**6/240 + t**7/90 + O(t**8)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = exp(sin(t))\n",
    "f.series(t, 0, 8)"
   ]
  },
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   "source": [
    "Output of the mathematical expression in LATEX typesetting is also possible"
   ]
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   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 + t + \\frac{t^{2}}{2} - \\frac{t^{4}}{8} - \\frac{t^{5}}{15} - \\frac{t^{6}}{240} + \\mathcal{O}\\left(t^{7}\\right)\n"
     ]
    }
   ],
   "source": [
    "from sympy import latex\n",
    "print latex(f.series(t, 0, 7))"
   ]
  },
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   "source": [
    "Using the above output, I will create this expression \n",
    "$$1 + t + \\frac{t^{2}}{2} - \\frac{t^{4}}{8} - \\frac{t^{5}}{15} - \\frac{t^{6}}{240} + \\mathcal{O}\\left(t^{7}\\right)$$"
   ]
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   "source": [
    "Lastly, we should also see tools for expanding and simplifying"
   ]
  },
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   "cell_type": "code",
   "execution_count": 13,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x + y)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import simplify, expand\n",
    "x, y = symbols('x y')\n",
    "f = -sin(x)*sin(y) + cos(x)*cos(y)\n",
    "simplify(f)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 14,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x + y)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(cos(x+y)) #Without any trignometric hint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "-sin(x)*sin(y) + cos(x)*cos(y)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(cos(x+y), trig=True) # With some trignometric hint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "a**2 + 2*a*b + b**2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a,b = symbols('a, b')\n",
    "f = (a+b)**2\n",
    "expand(f)"
   ]
  },
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   "execution_count": null,
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    "collapsed": true,
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