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  {
   "cell_type": "markdown",
   "id": "f03132be",
   "metadata": {},
   "source": [
    "# オイラーの公式\n",
    "\n",
    "オイラーの公式は、指数関数と円運動を結び付けます。\n",
    "\n",
    "$$\n",
    "e^{ix}=\\cos x+i\\sin x.\n",
    "$$\n",
    "\n",
    "この恒等式を係数ごとに確かめます。Egisonで記号式を微分し、\n",
    "両辺のマクローリン展開の各項を生成します。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9700a21b",
   "metadata": {},
   "source": [
    "## 指数関数のマクローリン展開\n",
    "\n",
    "関数$F$を原点のまわりで展開したとき、$n$次の項は\n",
    "\n",
    "$$\n",
    "\\frac{F^{(n)}(0)}{n!}x^n.\n",
    "$$\n",
    "\n",
    "まず、$e^{ix}$の最初の8項を求めます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ea4aa1af",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x : MathValue\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7544fc26",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{1, i x, \\frac{-1}{2} x^{2}, \\frac{-1}{6} i x^{3}, \\frac{1}{24} x^{4}, \\frac{1}{120} i x^{5}, \\frac{-1}{720} x^{6}, \\frac{-1}{5040} i x^{7}\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 8 (taylorExpansion (e^(i * x)) x 0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3b301e54",
   "metadata": {},
   "source": [
    "## 実部と虚部\n",
    "\n",
    "偶数次の項は$\\cos x$に、奇数次の項は$i\\sin x$に現れます。\n",
    "2つの級数を別々に計算することで、この偶奇による分かれ方が\n",
    "明確になります。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b930f243",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{1, 0, \\frac{-1}{2} x^{2}, 0, \\frac{1}{24} x^{4}, 0, \\frac{-1}{720} x^{6}, 0\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 8 (taylorExpansion (cos x) x 0)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e8ad66b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{0, i x, 0, \\frac{-1}{6} i x^{3}, 0, \\frac{1}{120} i x^{5}, 0, \\frac{-1}{5040} i x^{7}\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 8 (taylorExpansion (i * sin x) x 0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "429b78c9",
   "metadata": {},
   "source": [
    "## 係数ごとの比較\n",
    "\n",
    "2つのリストを加えると、表示したすべての次数について\n",
    "指数関数の級数が再現されるはずです。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "507f9da6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{1, i x, \\frac{-1}{2} x^{2}, \\frac{-1}{6} i x^{3}, \\frac{1}{24} x^{4}, \\frac{1}{120} i x^{5}, \\frac{-1}{720} x^{6}, \\frac{-1}{5040} i x^{7}\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 8\n",
    "  (map2 (+)\n",
    "    (taylorExpansion (cos x) x 0)\n",
    "    (taylorExpansion (i * sin x) x 0))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef4e7f8a",
   "metadata": {},
   "source": [
    "最後のリストは$e^{ix}$の展開と一致します。この計算は、$x$に\n",
    "数値を代入することなく、オイラーの公式が形式的冪級数の\n",
    "恒等式であることを示しています。\n"
   ]
  }
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