{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equação de Euler\n",
    "\n",
    "Euler introduziu o uso da função exponencial e logaritmos em provas analíticas. Ele descobriu maneiras de expressar várias funções logarítmicas usando séries de potências e definiu logaritmos com sucesso para números negativos e complexos, expandindo muito o escopo das aplicações matemáticas dos logaritmos. Ele também definiu a função exponencial para números complexos e descobriu sua relação com as funções trigonométricas.\n",
    "\n",
    "![img](https://raw.githubusercontent.com/the-akira/Python-Matematica/master/imagens/timeline.png)\n",
    "\n",
    "A Equação de Euler, também conhecida como identidade de Euler, é frequentemente citada como um exemplo de profunda beleza matemática.\n",
    "\n",
    "Ela é representada pela igualdade:\n",
    "\n",
    "\\begin{equation} e^{i\\pi} + 1 = 0\\end{equation}\n",
    "\n",
    "Em que:\n",
    "\n",
    "- $e$ é o número de Euler, a base dos logaritmos naturais\n",
    "- $i$ é a unidade imaginária, que satisfaz $i^2 = -1$\n",
    "- $\\pi$ é o conhecido pi, a razão entre a circunferência de um círculo e seu diâmetro\n",
    "\n",
    "Na Equação de Euler, três das operações aritméticas básicas ocorrem exatamente uma vez cada: **adição**, **multiplicação** e **exponenciação**. A identidade também vincula cinco constantes matemáticas fundamentais:\n",
    "\n",
    "- O número 0\n",
    "- O número 1\n",
    "- O número $\\pi$ ($\\pi = 3.1415...$)\n",
    "- O número $e$ ($e = 2.7182...$)\n",
    "- O número $i$, a unidade imaginária de números complexos\n",
    "\n",
    "Fundamentalmente a Identidade de Euler afirma que $e^{i \\pi}$ é igual a $-1$. A expressão $e^{i \\pi}$ é um caso especial da expressão $e^z$, onde $z$ pode ser qualquer número complexo. Em geral, $e^z$ é definido para o complexo $z$ estendendo uma das definições da função exponencial de expoentes reais para expoentes complexos. \n",
    "\n",
    "Por exemplo, uma definição comum é\n",
    "\n",
    "\\begin{equation} e^z = \\lim_{n\\to\\infty}\\left(1 + \\frac{{z}}{{n}}\\right)^n\\end{equation}\n",
    "\n",
    "A Identidade de Euler então afirma que o limite, ao $n$ aproximar o **infinito**, de $(1 + \\frac{i \\pi}{n})^n$ é igual a $-1$.\n",
    "\n",
    "Para compreendermos melhor essa ideia do limite, podemos observar a seguinte animação:\n",
    "\n",
    "![img](https://raw.githubusercontent.com/the-akira/Python-Matematica/master/imagens/ExpIPi.gif)\n",
    "\n",
    "A Identidade de Euler é considerada um caso especial da fórmula de Euler, ao qual afirma que para qualquer número real $x$:\n",
    "\n",
    "\\begin{equation} e^{i\\theta} = \\cos{\\theta} + i\\sin{\\theta}\\end{equation}\n",
    "\n",
    "Onde os dados de entrada das funções trigonométricas **seno** e **cosseno** são dados em radianos. Em particular, quando $x = \\pi$:\n",
    "\n",
    "\\begin{equation} e^{i\\theta} = \\cos{\\pi} + i\\sin{\\pi}\\end{equation}\n",
    "\n",
    "Sabemos que:\n",
    "\n",
    "\\begin{equation}\\cos{\\pi} = -1\\end{equation} \n",
    "\n",
    "\\begin{equation}\\sin{\\pi} = 0\\end{equation}\n",
    "\n",
    "Então temos que:\n",
    "\n",
    "\\begin{equation}e^{i\\pi} = -1 + 0i\\end{equation}\n",
    "\n",
    "Que nos leva até a Identidade de Euler:\n",
    "\n",
    "\\begin{equation} e^{i\\pi} + 1 = 0\\end{equation}\n",
    "\n",
    "Ao qual nos mostra a surpreendente conexão entre cinco dos mais significantes números da Matemática. \n",
    "\n",
    "Existem muitas provas para a **Identidade de Euler**, neste caso específico utilizaremos a prova com a **[série de Taylor](http://mathworld.wolfram.com/TaylorSeries.html)**, com a biblioteca **SymPy**.\n",
    "\n",
    "A ideia básica da série de Taylor é que muitas (porém, nem todas) funções na linha real podem ser representadas por polinômios de grau infinito. O mais conhecido exemplo de uma série de Taylor é a soma da série geométrica:\n",
    "\n",
    "\\begin{equation}\\frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + x^5 ...\\end{equation}\n",
    "\n",
    "para $|x| < 1$\n",
    "\n",
    "Podemos ilustrar essa soma em SymPy da seguinte maneira:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4    5    ⎛ 6⎞\n",
       "1 + x + x  + x  + x  + x  + O⎝x ⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "\n",
    "x = sym.Symbol('x')\n",
    "sym.series(1 / (1 - x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O termo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       " ⎛ 6⎞\n",
       "O⎝x ⎠"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Order(x**6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ele indica que todos os termos omitidos possuem um grau maior que ou igual a 6. Naturalmente, SymPy pode apenas calcular muitos termos finitos de funções de série de Taylor. Nós podemos controlar o número de termos da série de Taylor calculada por SymPy ao passarmos o argumento opcional **n** para sympy.series. Por exemplo, nós podemos calcular os 10 primeiros termos da série geométrica da seguinte maneira:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6} + x^{7} + x^{8} + x^{9} + O\\left(x^{10}\\right)$"
      ],
      "text/plain": [
       "         2    3    4    5    6    7    8    9    ⎛ 10⎞\n",
       "1 + x + x  + x  + x  + x  + x  + x  + x  + x  + O⎝x  ⎠"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.series(1 / (1 - x), n=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nós agora alteramos nossa atenção para a Fórmula de Euler ao definirmos a variável $\\theta$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAATCAYAAACgADyUAAAABHNCSVQICAgIfAhkiAAAASNJREFUOI21078rRlEYwPHPK968SYyvFP4AWQwMDEopJhMjZWEmKekdZJUyGZTlTRaDZDIyML0ljEJIDGTwYzHco27XvZTy1Onbec75nvOcnntzpVLJX6IqI1/AAs7xiissoeYnsQnHmEcFK3jGHFazxDx20IY+jGAWXbjEBIpp4jQ6w+aDWP4F22F/b1IsYAa3WEt5wmPgtxuH0YgyPlLE2sB3qI4tDAU2o5Qi9gdeJcWewNEUKR6n8VLr0IIT5FJGfSj/GhdxsTnwJuOWAVHzd78SX2I+8C1DHA9cT4p3gcUUqRuD2MNRUnzAmaj5HTGpFZt4wlT8tHgfF8N8H8uij6CCBlGrLrLEMsZwj8lQ3hbacZisvzox3wjj18j6H/9P/AS7szQCO9fAVQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\theta$"
      ],
      "text/plain": [
       "\\theta"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta = sym.Symbol('\\\\theta', real=True)\n",
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Também definimos funções para calcularmos a série de Taylor de $\\sin{\\theta}$ e $\\cos{\\theta}$ para qualquer grau de **n**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sin_series(n):\n",
    "    return sym.series(sym.sin(theta), n=n)\n",
    "\n",
    "def cos_series(n):\n",
    "    return sym.series(sym.cos(theta), n=n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Os termos iniciais para a série de $\\sin{\\theta}$ são:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\theta - \\frac{\\theta^{3}}{6} + \\frac{\\theta^{5}}{120} - \\frac{\\theta^{7}}{5040} + \\frac{\\theta^{9}}{362880} + O\\left(\\theta^{10}\\right)$"
      ],
      "text/plain": [
       "               3         5         7         9              \n",
       "         \\theta    \\theta    \\theta    \\theta     ⎛      10⎞\n",
       "\\theta - ─────── + ─────── - ─────── + ─────── + O⎝\\theta  ⎠\n",
       "            6        120       5040     362880              "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 10\n",
    "sin_series(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Os termos iniciais para a série de $\\cos{\\theta}$ são:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - \\frac{\\theta^{2}}{2} + \\frac{\\theta^{4}}{24} - \\frac{\\theta^{6}}{720} + \\frac{\\theta^{8}}{40320} + O\\left(\\theta^{10}\\right)$"
      ],
      "text/plain": [
       "          2         4         6         8              \n",
       "    \\theta    \\theta    \\theta    \\theta     ⎛      10⎞\n",
       "1 - ─────── + ─────── - ─────── + ─────── + O⎝\\theta  ⎠\n",
       "       2         24       720      40320               "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos_series(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos comparar essas duas séries com $e^{i\\theta}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + i \\theta - \\frac{\\theta^{2}}{2} - \\frac{i \\theta^{3}}{6} + \\frac{\\theta^{4}}{24} + \\frac{i \\theta^{5}}{120} - \\frac{\\theta^{6}}{720} - \\frac{i \\theta^{7}}{5040} + \\frac{\\theta^{8}}{40320} + \\frac{i \\theta^{9}}{362880} + O\\left(\\theta^{10}\\right)$"
      ],
      "text/plain": [
       "                     2           3         4           5         6           7\n",
       "               \\theta    ⅈ⋅\\theta    \\theta    ⅈ⋅\\theta    \\theta    ⅈ⋅\\theta \n",
       "1 + ⅈ⋅\\theta - ─────── - ───────── + ─────── + ───────── - ─────── - ─────────\n",
       "                  2          6          24        120        720        5040  \n",
       "\n",
       "         8           9              \n",
       "   \\theta    ⅈ⋅\\theta     ⎛      10⎞\n",
       " + ─────── + ───────── + O⎝\\theta  ⎠\n",
       "    40320      362880               "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def exp_series(n):\n",
    "    return sym.series(sym.exp(sym.I * theta), n=n)\n",
    "\n",
    "exp_series(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A ideia chave por trás dessa prova da Fórmula de Euler é que os termos dessa soma contendo $i$ são idênticos (à parte de $i$) ao termos da série de Taylor para $\\sin{\\theta}$. Similarmente, os termos não contendo $i$ são idênticos aos termos da série de Taylor de $\\cos{\\theta}$.\n",
    "\n",
    "Nós podemos ver isso com mais clareza ao pedirmos para SymPy coletar os termos contendo $i$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + i \\left(\\theta - \\frac{\\theta^{3}}{6} + \\frac{\\theta^{5}}{120} - \\frac{\\theta^{7}}{5040} + \\frac{\\theta^{9}}{362880} + O\\left(\\theta^{10}\\right)\\right) - \\frac{\\theta^{2}}{2} + \\frac{\\theta^{4}}{24} - \\frac{\\theta^{6}}{720} + \\frac{\\theta^{8}}{40320} + O\\left(\\theta^{10}\\right)$"
      ],
      "text/plain": [
       "      ⎛               3         5         7         9              ⎞         2\n",
       "      ⎜         \\theta    \\theta    \\theta    \\theta     ⎛      10⎞⎟   \\theta \n",
       "1 + ⅈ⋅⎜\\theta - ─────── + ─────── - ─────── + ─────── + O⎝\\theta  ⎠⎟ - ───────\n",
       "      ⎝            6        120       5040     362880              ⎠      2   \n",
       "\n",
       "         4         6         8              \n",
       "   \\theta    \\theta    \\theta     ⎛      10⎞\n",
       " + ─────── - ─────── + ─────── + O⎝\\theta  ⎠\n",
       "      24       720      40320               "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.collect(exp_series(n), sym.I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imediatamente podemos reconhecer essa expresão como $\\cos{\\theta} + i\\sin{\\theta}$.\n",
    "\n",
    "Além dessa inspeção visual, SymPy torna ligeiramente fácil a verificação se a série de Taylor está de acordo com quantidade grande de número de termos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAYCAYAAAC2odCOAAAABHNCSVQICAgIfAhkiAAABC9JREFUWIXt2GmIllUUB/DfqInTCAraYkJZiUbRhka2CYFWZB8S2yizsiKKkghLg4pJzILIIQgLI8k2KrASKVGQwjAiK5NEK4IyUdNs0hZSs+zDuY8+Pt73nZncJvAPD+d9zzl3O/cs996G5uZmh1EfXQ71BNrAcMzDOuzEVTX07sJ32IrPcNF/1MmisxupCctxdx2da/E0puFsLMF8HN9BnZo4EEZ6CRvFAnNoxCP4SuzqGjH5IzK68/EQ3qoz3n14Ec9jFSZgPe7soM4Q4a23VgeoZ6ST8BiWoRXb8H0a7MwabYZiLJ7AHxl5PywVC18udvdXPIhn6sylFrqLxS2s8Bfi/A7oECH4DqaiZ1kxZ6QGPIyVYvKb8SpaxO6Pw6cYn2k7TSz62RoLmocBuFiEwCScix9wG47NtKuHvuiKDRX+hlJf7dEp8HjiTSgzq0ZqwCxMwZc4RSzoHkzGZRiZdGeK+C4wCCPwJv7MLGii2NFJIicU+B1vp7m0O5lWsDOzjiqvPTqfCEe4QxgWextpMm62O/t/k5nQIuEpXXFviT8+DfxGpk0j7hd5YGZG/nOiHfWkTfg70+5ouz2nPTplvC4S+oiCUTbSiXhUJNOrE62FBYmeV+KNSJP5OKM/Gr3xGv7KyHskur3OmDlsFxs6ssIfiY86oFPGkpIcdCsJJ4oKM0OcJ+phTaK9Em3CWaJy5BL2qET7ozkjL3ZtTYXfEwNL/wekcVpFHoPpeFmEyhIRKsfhuVK79ugUWJro8IJRGKlB7Da8kmlYRZ9ENyXaX4Tf+hr6FyZ6XRv9rqz8H4r3S/+fTHS2SAtEePcRFbMfVuByrC61a49OgS0iinadoQoj9UvfDnzRxkJgWKKfJ1oY7ZeMblMacAVOz8h7Cs/YII4YZXwgNrAtzEjfvuoUaMUxxZ8iJ/VN9DdhqHpowA3pd3HIK6pZj73V9U90XY3+LhVh/m4b4x5MNCpV6MJImxPtjSPb6OB6nCbyz9zE25hon4x+90S31ejvlkRntTHuwUIXYYeNZQaRBNcKL6lWgTIGCZfdIQ5//yT+evyEwZk2PyaaK+/DRG6YL5JqZ8BgYYddaad8BGhJdLrI/FVcISpDE263Z/ncicUibAdW2m0SXjcEZ5T4J4gzyRZxQ+8sKPLtroJRPgK0iPI6Vpw65woPOwoX4FThFaPF9aKKORgjcsy3FdlUcbVZJKpnE64Rxh1l74R9KHGJOO8VqURD5tFtjLgJnyNisxVfi6vDC+JulkN3YdTV4j5WxU14ACcL73pPHF7X/peVHCD0Eo6wAFcWzG4ZxTnp6yi22/PNZllFPjt9nRnjRIV+qszc3+9JLcKbpuznfg8GGsWrxxx8WBbsbyNtxY3iKaXWo1tnxQBx+Z5YFeTCbV+xOH3/N6ySv1d2+jfuToHDRmoH/gWgWfc2gW2h3gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle O\\left(\\theta^{100}\\right)$"
      ],
      "text/plain": [
       " ⎛      100⎞\n",
       "O⎝\\theta   ⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 100\n",
    "\n",
    "sym.simplify(exp_series(N) - (cos_series(N) + sym.I * sin_series(N)))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}