{"id":33014,"date":"2023-11-02T22:30:11","date_gmt":"2023-11-02T22:30:11","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%85%AC%E5%BC%8F-%E5%AE%9A%E7%BE%A9\/"},"modified":"2023-11-02T22:30:11","modified_gmt":"2023-11-02T22:30:11","slug":"%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%85%AC%E5%BC%8F-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=33014","title":{"rendered":"\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f – \u5b9a\u7fa9"},"content":{"rendered":"
\n\"\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\"<\/figure>
\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f
$$ {\\mathrm e^{\\mathrm i\\varphi}=\\cos\\varphi+\\mathrm i\\sin\\varphi} $$<\/div><\/div><\/div><\/div>

\u30b9\u30a4\u30b9\u306e\u6570\u5b66\u8005\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30fb\u30aa\u30a4\u30e9\u30fc\u306b\u3088\u308b\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u306f<\/strong>\u6b21\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u3066\u3044\u307e\u3059\u3002<\/p>

\u4efb\u610f\u306e\u5b9f\u6570x<\/i>\u306b\u5bfe\u3057\u3066\u3001
$$ {e^{ix} = \\cos x + i\\;\\sin x} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u3001 e<\/i>\u306f\u5bfe\u6570\u306e\u81ea\u7136\u5e95\u3001 i \u306f<\/i>\u865a\u6570<\/a><\/span>\u3001sin \u3068 cos \u306f\u4e09\u89d2\u95a2\u6570\u3067\u3059\u3002<\/p>

\u8aac\u660e<\/span><\/h2>

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$$ {x\\mapsto e^{ix}} $$<\/div>\u306f\u3001 x \u304c<\/i>\u5b9f\u6570\u306e\u96c6\u5408<\/a><\/span>\u5185\u3067\u5909\u5316\u3059\u308b\u3068\u304d\u306e\u8907\u7d20\u5e73\u9762\u5185\u306e\u5358\u4f4d\u5186\u3092\u8868\u3057\u307e\u3059\u3002 x \u306f<\/i><\/span>\u3001\u539f\u70b9\u306e\u7aef\u306b\u3042\u308b\u534a\u7dda<\/a><\/span>\u3068\u3001\u6b63\u306e\u5b9f\u6570\u306e\u534a\u7dda\u3067\u5358\u4f4d\u5186\u306e\u70b9<\/a><\/span>\u3092\u901a\u904e\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u4f5c\u3089\u308c\u308b\u65b9\u5411\u89d2<\/a><\/span>\u306e\u5c3a\u5ea6\u3092\u8868\u3057\u307e\u3059\u3002\u3053\u306e\u5f0f\u306f\u3001sin \u3068 cos \u306e\u5f15\u6570\u304c\u5ea6\u3067\u306f\u306a\u304f\u30e9\u30b8\u30a2\u30f3\u3067\u8868\u73fe\u3055\u308c\u3066\u3044\u308b\u5834\u5408\u306b\u306e\u307f\u6709\u52b9\u3067\u3059\u3002<\/p>

\u8a3c\u660e\u306f<\/a><\/span>\u6307\u6570<\/a><\/span>\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u7d1a\u6570\u5c55\u958b\u306b\u57fa\u3065\u3044\u3066\u3044\u307e\u3059

$$ {z\\mapsto e^z} $$<\/div>\u8907\u7d20\u5909\u6570z<\/i>\u3068\u5b9f\u6570\u5909\u6570\u3068\u307f\u306a\u3055\u308c\u308b\u95a2\u6570 sin \u304a\u3088\u3073 cos \u306e\u5024\u3002\u5b9f\u969b\u3001\u540c\u3058\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304c\u3059\u3079\u3066\u306e\u8907\u7d20\u6570x<\/i>\u306b\u5bfe\u3057\u3066\u4f9d\u7136\u3068\u3057\u3066\u6709\u52b9\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

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$$ {e^{a + b} = e^a \\cdot e^{b}} $$<\/div><\/dd><\/dl>

\u305d\u3057\u3066<\/p>

$$ {(e^a)^b = e^{a b} \\,} $$<\/div><\/dd><\/dl>

(\u3053\u308c\u3089\u306f\u3059\u3079\u3066\u306e\u8907\u7d20\u6570a<\/i><\/span>\u304a\u3088\u3073b<\/i><\/span>\u306b\u3082\u5f53\u3066\u306f\u307e\u308a\u307e\u3059)\u3001\u3044\u304f\u3064\u304b\u306e\u4e09\u89d2\u6052\u7b49\u5f0f\u3092\u5c0e\u304d\u51fa\u3057\u305f\u308a\u3001\u30c9 \u30e2\u30a2\u30d6\u30eb\u306e\u516c\u5f0f<\/a><\/span>\u3092\u63a8\u5b9a\u3057\u305f\u308a\u3059\u308b\u3053\u3068\u304c\u7c21\u5358\u306b\u306a\u308a\u307e\u3059\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3067\u306f\u3001\u30b3\u30b5\u30a4\u30f3<\/a><\/span>\u95a2\u6570\u3068\u30b5\u30a4\u30f3<\/a><\/span>\u95a2\u6570\u3092\u6307\u6570\u95a2\u6570\u306e\u552f\u4e00\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u3068\u3057\u3066\u89e3\u91c8\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {\\cos x = {e^{ix} + e^{-ix} \\over 2}} $$<\/div><\/dd>
$$ {\\sin x = {e^{ix} – e^{-ix} \\over 2i}} $$<\/div><\/dd><\/dl>

\u3053\u308c\u3089\u306e\u516c\u5f0f (\u30aa\u30a4\u30e9\u30fc\u516c\u5f0f\u3068\u3082\u547c\u3070\u308c\u307e\u3059) \u306f\u3001\u8907\u7d20\u5909\u6570x<\/i><\/span>\u306e\u4e09\u89d2\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u3057\u3066\u6a5f\u80fd\u3057\u307e\u3059\u3002\u305d\u308c\u3089\u3092\u53d6\u5f97\u3059\u308b\u306b\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u5c0e\u304d\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {e^{ix} = \\cos x + i \\sin x \\,} $$<\/div><\/dd>
$$ {e^{-ix} = \\cos x – i \\sin x \\,} $$<\/div><\/dd><\/dl>

\u305d\u3057\u3066\u30b3\u30b5\u30a4\u30f3\u304b\u30b5\u30a4\u30f3\u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002<\/p>

\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u306f\u3001\u95a2\u6570\u306f

$$ {x\\mapsto e^{ix}} $$<\/div> \u3001\u30b5\u30a4\u30f3\u3068\u30b3\u30b5\u30a4\u30f3\u3092\u4f7f\u7528\u3057\u3066\u8868\u73fe\u3055\u308c\u308b\u5b9f\u969b\u306e\u89e3\u3092\u6c7a\u5b9a\u3059\u308b\u3053\u3068\u304c\u554f\u984c\u3067\u3059\u304c\u3001\u5c0e\u51fa\u3092\u5358\u7d14\u5316\u3059\u308b\u305f\u3081\u306b\u3088\u304f\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u6052\u7b49\u5f0f\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u306e\u76f4\u63a5\u306e\u7d50\u679c\u3067\u3059\u3002<\/p>

\u96fb\u6c17\u5de5\u5b66<\/span>\u3084\u305d\u306e\u4ed6\u306e\u5206\u91ce\u3067\u306f\u3001\u6642\u9593<\/a><\/span>\u3068\u3068\u3082\u306b\u5468\u671f\u7684\u306b\u5909\u5316\u3059\u308b\u4fe1\u53f7\u306f\u3001\u30b5\u30a4\u30f3\u95a2\u6570\u3068\u30b3\u30b5\u30a4\u30f3\u95a2\u6570\u306e\u7dda\u5f62\u7d50\u5408\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\u3053\u3068\u304c\u591a\u304f (\u30d5\u30fc\u30ea\u30a8\u89e3\u6790\u3092\u53c2\u7167)\u3001\u5f8c\u8005\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306e\u95a2\u6570\u3092\u4f7f\u7528\u3057\u3066\u3001\u865a\u6570\u6307\u6570\u3092\u542b\u3080\u6307\u6570\u95a2\u6570\u306e\u5b9f\u90e8\u3068\u3057\u3066\u8868\u73fe\u3059\u308b\u65b9\u304c\u4fbf\u5229\u3067\u3059\u3002\u5f0f\u3002<\/p>

\n\"\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/figure>

\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3<\/span><\/h2>

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\u5b9f\u5909\u6570x<\/i>\u306e\u95a2\u6570exp<\/i>\u306e\u7d1a\u6570\u5c55\u958b\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {e^x = \\frac{x^0}{0!} + \\frac{x^1}{1!} + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + … = \\sum_{n=0}^\\infty \\frac{x^n}{n!}} $$<\/div><\/dd><\/dl>

\u305d\u3057\u3066\u4efb\u610f\u306e<\/a><\/span>\u8907\u7d20\u6570<\/a><\/span>x<\/i>\u306b\u62e1\u5f35\u3055\u308c\u307e\u3059\u3002<\/p>

i \u3092<\/i>\u6307\u6570<\/a><\/span>\u306b\u6ce8\u5165\u3059\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {e^{ix} = \\sum_{n=0}^\\infty \\frac{{(ix)}^n}{n!} = \\sum_{n=0}^\\infty \\frac{i^n x^n}{n!} = \\sum_{n=0}^\\infty \\frac{x^n}{n!} i^n} $$<\/div><\/dd><\/dl>

\u3053\u306e\u9000\u5ec3\u7684\u306a\u6587\u7ae0\u3092\u5f97\u308b\u305f\u3081\u306b\u3001\u305d\u306e\u7528\u8a9e\u3092\u30b0\u30eb\u30fc\u30d7\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {e^{ix} = \\sum_{n=0}^\\infty \\left( \\frac{x^{4n}} {(4n)!} i^{\\,4n} + \\frac{x^{4n+1}}{(4n+1)!} i^{\\,4n+1} + \\frac{x^{4n+2}}{(4n+2)!} i^{\\,4n+2} + \\frac{x^{4n+3}}{(4n+3)!} i^{\\,4n+3} \\right)} $$<\/div><\/dd><\/dl>

\u3053\u308c\u3092\u5358\u7d14\u5316\u3059\u308b\u305f\u3081\u306b\u3001\u6b21\u306ei<\/i>\u306e\u57fa\u672c\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002 <\/p>

$$ {i^0 = 1, \\qquad i^1 = i, \\qquad i^2 = -1, \\qquad i^3 = -i, \\qquad i^4 = 1, \\ldots} $$<\/div><\/dd><\/dl>

\u4efb\u610f\u306e\u6574\u6570\u6307\u6570\u306b\u4e00\u822c\u5316\u3059\u308b\u3068\u3001\u3059\u3079\u3066\u306en<\/i>\u306b\u5bfe\u3057\u3066\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {i^{\\,4n} = 1, \\qquad i^{\\,4n+1} = i, \\qquad i^{\\,4n+2} = -1, \\qquad i^{\\,4n+3} = -i} $$<\/div><\/dd><\/dl>

\u305d\u308c\u3067\u3001 <\/p>

$$ {e^{ix} = \\sum_{n=0}^\\infty \\left( \\frac{x^{4n}} {(4n)!} + \\frac{x^{4n+1}}{(4n+1)!} i – \\frac{x^{4n+2}}{(4n+2)!} – \\frac{x^{4n+3}}{(4n+3)!} i \\right)} $$<\/div><\/dd><\/dl>

\u9805\u3092\u4e26\u3079\u66ff\u3048\u3066\u5408\u8a08\u3092 2 \u3064\u306b\u5206\u5272\u3057\u307e\u3059 (2 \u3064\u306e\u7cfb\u5217\u304c\u7d76\u5bfe\u306b\u53ce\u675f\u3059\u308b\u305f\u3081\u3001\u3053\u308c\u304c\u53ef\u80fd\u3067\u3059)\u3002 <\/p>

$$ {e^{ix} = \\sum_{n=0}^\\infty \\left( \\frac{x^{4n}} {(4n)!} – \\frac{x^{4n+2}}{(4n+2)!} \\right) + i\\,\\sum_{n=0}^\\infty \\left( \\frac{x^{4n+1}}{(4n+1)!} – \\frac{x^{4n+3}}{(4n+3)!} \\right)} $$<\/div><\/dd><\/dl>

\u3082\u3046\u5c11\u3057\u5148\u306b\u9032\u3080\u305f\u3081\u306b\u3001\u30b3\u30b5\u30a4\u30f3\u95a2\u6570\u3068\u30b5\u30a4\u30f3\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u7d1a\u6570\u5c55\u958b\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002 <\/p>

$$ {\\cos x = 1 – \\frac{x^2}{2!} + \\frac{x^4}{4!} – \\frac{x^6}{6!} + … = \\sum_{n=0}^\\infty \\left( \\frac{x^{4n}} {(4n)!} – \\frac{x^{4n+2}}{(4n+2)!} \\right)} $$<\/div><\/dd><\/dl>
$$ {\\sin x = x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} + … = \\sum_{n=0}^\\infty \\left( \\frac{x^{4n+1}}{(4n+1)!} – \\frac{x^{4n+3}}{(4n+3)!} \\right)} $$<\/div><\/dd><\/dl>

\u524d\u306e\u5f0f\u306ee<\/i> ix \u3092<\/i><\/sup>\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {e^{ix} = \\cos x + i\\; \\sin x} $$<\/div><\/dd><\/dl>

\u5fc5\u8981\u306b\u5fdc\u3058\u3066\u3002<\/p>

\u3053\u306e\u5225\u306e\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3\u3067\u306f\u5fae\u5206<\/a><\/span>\u7a4d\u5206\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>

\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u3092\u5b9a\u7fa9\u3057\u307e\u3057\u3087\u3046

$$ {f \\} $$<\/div>\u306b\u3088\u308b<\/p>
$$ {f(x) = \\frac{\\cos x+i\\sin x}{e^{ix}}. \\} $$<\/div><\/dd><\/dl>

\u3053\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u306f\u660e\u78ba\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u305f\u3081\u3001 <\/p>

$$ {e^{ix}\\cdot e^{-ix}=e^0=1 \\} $$<\/div><\/dd><\/dl>

\u3068\u3044\u3046\u3053\u3068\u3092\u6697\u793a\u3057\u307e\u3059

$$ {e^{ix} \\} $$<\/div>\u6c7a\u3057\u3066\u30bc\u30ed\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>

\u30a2\u30d7\u30ea

$$ {f \\} $$<\/div>\u306f 2 \u3064\u306e\u5fae\u5206\u53ef\u80fd\u306a\u95a2\u6570\u306e\u5546\u3067\u3042\u308b\u305f\u3081\u3001\u5fae\u5206\u53ef\u80fd (\u5546\u306e\u5c0e\u51fa) \u3067\u3042\u308a\u3001\u305d\u306e\u5c0e\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u307e\u3059\u3002 <\/p>
$$ {f'(x)\\,} $$<\/div><\/td>
$$ {= \\displaystyle\\frac{(-\\sin x+i\\cos x)\\cdot e^{ix} – (\\cos x+i\\sin x)\\cdot i\\cdot e^{ix}}{(e^{ix})^2} \\} $$<\/div><\/td><\/tr>
$$ {= \\displaystyle\\frac{-\\sin x\\cdot e^{ix}-i^2\\sin x\\cdot e^{ix}}{(e^{ix})^2} \\} $$<\/div><\/td><\/tr>
$$ {= \\displaystyle\\frac{-\\sin x-i^2\\sin x}{e^{ix}} \\} $$<\/div><\/td><\/tr>
$$ {= \\displaystyle\\frac{-\\sin x-(-1)\\sin x}{e^{ix}} \\} $$<\/div><\/td><\/tr>
$$ {= \\displaystyle\\frac{-\\sin x+\\sin x}{e^{ix}} \\} $$<\/div><\/td><\/tr>
$$ {= 0 \\} $$<\/div><\/td><\/tr><\/table><\/dd><\/dl>

\u305d\u308c\u3067\u3001

$$ {f \\} $$<\/div>\u306f\u5b9a\u6570\u95a2\u6570\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001 <\/p>
$$ {f(x)=f(0)=\\frac{\\cos 0 + i \\sin 0}{e^0}=1} $$<\/div><\/td><\/tr>
$$ {\\frac{\\cos x + i \\sin x}{e^{ix}}=1} $$<\/div><\/td><\/tr>
$$ {\\displaystyle\\cos x + i \\sin x=e^{ix}} $$<\/div><\/td><\/tr><\/table><\/dd><\/dl>

\u6b74\u53f2\u7684<\/span><\/h2>

\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u306f\u30011714 \u5e74\u306b\u30ed\u30b8\u30e3\u30fc \u30b3\u30fc\u30c4\u306b\u3088\u3063\u3066 ln(cos( x<\/i> ) + i<\/i> sin( x<\/i> )) = ix<\/i> (ln \u306f\u81ea\u7136\u5bfe\u6570\u3001\u3064\u307e\u308a Log Basic e \u3092\u8868\u3057\u307e\u3059) \u3068\u3044\u3046\u5f62\u5f0f\u3067\u6700\u521d\u306b\u5b9f\u8a3c\u3055\u308c\u307e\u3057\u305f[<\/span> 1 ]<\/span><\/sup> \u3002 2 \u3064\u306e\u7d1a\u6570\u9593\u306e\u7b49\u4fa1\u6027\u306e\u8a3c\u660e\u306b\u57fa\u3065\u3044\u3066\u30011748 \u5e74\u306b\u73fe\u5728\u306e\u5f62\u5f0f\u3067\u516c\u5f0f\u3092\u767a\u8868\u3057\u305f\u306e\u306f\u30aa\u30a4\u30e9\u30fc\u3067\u3057\u305f\u3002\u3069\u3061\u3089\u306e\u6570\u5b66\u8005\u3082\u3001\u5f0f\u306e\u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\u3092\u4e0e\u3048\u307e\u305b\u3093\u3067\u3057\u305f\u3002\u8907\u7d20\u6570\u3092\u5e73\u9762\u4e0a\u306e\u70b9\u3068\u3057\u3066\u89e3\u91c8\u3059\u308b\u3053\u3068\u306f\u300150 \u5e74\u5f8c\u307e\u3067\u5b9f\u969b\u306b\u8b70\u8ad6\u3055\u308c\u307e\u305b\u3093\u3067\u3057\u305f\u3002 (\u30ab\u30b9\u30d1\u30fc\u30fb\u30a6\u30a7\u30c3\u30bb\u30eb\u3092\u53c2\u7167)\u3002<\/p><\/div>

\n\"\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Euler se formule \u2013 afrikaans<\/a><\/li>
  2. \u0635\u064a\u063a\u0629 \u0623\u0648\u064a\u0644\u0631 \u2013 arabe<\/a><\/li>
  3. F\u00f3rmula d’Euler \u2013 asturien<\/a><\/li>
  4. Eyler d\u00fcsturu \u2013 azerba\u00efdjanais<\/a><\/li>
  5. \u042d\u0439\u043b\u0435\u0440 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u04bb\u044b \u2013 bachkir<\/a><\/li>
  6. \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u042d\u0439\u043b\u0435\u0440\u0430 \u2013 bi\u00e9lorusse<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n