{"id":6063,"date":"2023-04-06T11:53:16","date_gmt":"2023-04-06T02:53:16","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6063"},"modified":"2024-03-15T14:49:18","modified_gmt":"2024-03-15T05:49:18","slug":"sympy-%e3%81%a7%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b%e3%83%bb%e3%82%aa%e3%82%a4%e3%83%a9%e3%83%bc%e3%81%ae%e5%85%ac%e5%bc%8f","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sympy-%e3%81%a7%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b%e3%83%bb%e3%82%aa%e3%82%a4%e3%83%a9%e3%83%bc%e3%81%ae%e5%85%ac%e5%bc%8f\/","title":{"rendered":"SymPy \u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u30fb\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f"},"content":{"rendered":"

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Python \u3067\u5fae\u5206\u7a4d\u5206\u306a\u3069\u306e\u8a18\u53f7\u8a08\u7b97\u3092\u3059\u308b\u5834\u5408\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \u3057\u307e\u3059\u3002\uff08\u5f18\u5927 JupyterHub \u3067\u306f\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306f\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u6e08\u307f\u3067\u3059\u3002\uff09<\/p>\n

\u30bb\u30eb\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\uff0c\u4e0a\u306e\u30e1\u30cb\u30e5\u30fc\u304b\u3089\u300c\u25b6\ufe0e Run\u300d\u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u304b\uff0cShift \u30ad\u30fc\u3092\u62bc\u3057\u306a\u304c\u3089 Enter \u30ad\u30fc\uff08Return \u30ad\u30fc\uff09\u3092\u62bc\u3059\u3068\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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from sympy.abc<\/span> import *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9ad8\u968e\u5c0e\u95a2\u6570<\/h3>\n
$f(x)$ \u3092 $x$ \u3067 $n$ \u968e\u5fae\u5206\u3059\u308b\u4f8b\u3002SymPy \u3067\u306f diff(f(x), x, n)<\/code> \u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002
\n\u4f8b\uff1a<\/p>\n
$$\\frac{d^2}{dx^2} \\sin x = \\cdots$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u624b\u3063\u53d6\u308a\u65e9\u304f\u7b54\u3048\u3092\u77e5\u308a\u305f\u3044\u3068\u304d\u306f...<\/span>\r\n\r\ndiff<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
$\\displaystyle – \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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# \u4f55\u3092\u8a08\u7b97\u3059\u308b\u304b\u3092\u8868\u793a\u3055\u305b\u3066\u304b\u3089\u5b9f\u884c\u3059\u308b\u5834\u5408\u306f...<\/span>\r\n\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>),<\/span> \r\n   diff<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} \\sin{\\left(x \\right)} = – \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h3>\n
\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f series()<\/code> \u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u6307\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n
$f(x) = e^x$ \u306e $x = 0$ \u306e\u307e\u308f\u308a\u306b $x^5$ \u307e\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u4f8b\u30025\u6b21\u307e\u3067\u306e\u5c55\u958b\u306a\u306e\u306b\u306a\u305c 6<\/code> \u304b\u3068\u60a9\u3080\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u3053\u308c\u304c Python \u306e\u6d41\u5100\u3089\u3057\u3044\u3002<\/p>\n
$x^0$ \u304b\u3089\u6570\u3048\u3066 6 \u756a\u76ee\u307e\u3067\u3060\u304b\u3089 6<\/code> \u3068\u304b\uff0c5<\/code> \u306f 6<\/code> \u672a\u6e80\u306a\u306e\u3067 6<\/code> \u3068\u66f8\u304f\u3068\u304b\uff0c\u5404\u81ea\u7d0d\u5f97\u3059\u308b\u65b9\u6cd5\u3067\u899a\u3048\u3066\u304a\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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f<\/span> =<\/span> exp<\/span>(<\/span>x<\/span>)<\/span>\r\nseries<\/span>(<\/span>f<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 6<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$O(x^6)$ \u306f $x^6$ \u4ee5\u4e0a\u306e\u9ad8\u6b21\u306e\u9805\u3068\u3044\u3046\u610f\u5473\u3002<\/p>\n
\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u5f0f\u3067 $x=1$ \u3068\u304a\u304f\u3068
\n$$ e = \\sum_{k=0}^{\\infty} \\frac{1}{k!}$$ \u3068\u306a\u308a\uff0c\u3053\u306e\u7d1a\u6570\u5c55\u958b\u3092\u4f7f\u3063\u3066\u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u306e\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066\uff0c$\\displaystyle \\sum_{k=0}^{10} \\frac{1}{k!}$ \u306e\u8a08\u7b97\u3002SymPy \u3067\u306f\u968e\u4e57 $k!$ \u306f factorial(k)<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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# \u624b\u3063\u53d6\u308a\u65e9\u304f\u7dcf\u548c\u306e\u7b54\u3048\u3092\u77e5\u308a\u305f\u3044\u3068\u304d\u306f...<\/span>\r\n\r\nsummation<\/span>(<\/span>1<\/span>\/<\/span>factorial<\/span>(<\/span>k<\/span>),<\/span> (<\/span>k<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
$\\displaystyle \\frac{9864101}{3628800}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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# \u4f55\u3092\u8a08\u7b97\u3059\u308b\u304b\u3092\u8868\u793a\u3055\u305b\u3066\u304b\u3089\u5b9f\u884c\u3059\u308b\u5834\u5408\u306f...<\/span>\r\n\r\nEq<\/span>(<\/span>Sum<\/span>(<\/span>1<\/span>\/<\/span>factorial<\/span>(<\/span>k<\/span>),<\/span> (<\/span>k<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>)),<\/span> \r\n   Sum<\/span>(<\/span>1<\/span>\/<\/span>factorial<\/span>(<\/span>k<\/span>),<\/span> (<\/span>k<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\sum_{k=0}^{10} \\frac{1}{k!} = \\frac{9864101}{3628800}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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# \u6d6e\u52d5\u5c0f\u6570\u70b9\u6570\u3067\u8868\u793a<\/span>\r\n\r\nfloat<\/span>(<\/span>summation<\/span>(<\/span>1<\/span>\/<\/span>factorial<\/span>(<\/span>k<\/span>),<\/span> (<\/span>k<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
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2.7182818011463845<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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# \u30cd\u30a4\u30d4\u30a2\u6570\u3068\u6bd4\u8f03<\/span>\r\n\r\nfloat<\/span>(<\/span>E<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
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2.718281828459045<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e09\u89d2\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n
$f(x) = \\sin x$ \u306e $x = 0$ \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f
\n\\begin{eqnarray}
\n\\sin x &=& f(0) + f'(0)\\, x + \\frac{f”(0)}{2!}\\,x^2 + \\frac{f”'(0)}{3!}\\,x^3 +\\cdots \\\\
\n&=& \\sin 0 + \\cos 0 \\cdot x + \\frac{-\\sin 0}{2!}\\,x^2 + \\frac{-\\cos 0}{3!}\\,x^3 + \\cdots \\\\
\n&=& x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} + \\cdots\\\\
\n&=& \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)!}\\,x^{2n+1}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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series<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 8<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle x – \\frac{x^{3}}{6} + \\frac{x^{5}}{120} – \\frac{x^{7}}{5040} + O\\left(x^{8}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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# \u968e\u4e57\u306e\u78ba\u8a8d<\/span>\r\n\r\nfactorial<\/span>(<\/span>3<\/span>),<\/span> factorial<\/span>(<\/span>5<\/span>),<\/span> factorial<\/span>(<\/span>7<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
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(6, 120, 5040)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$f(x) = \\cos x$ \u306e $x = 0$ \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f
\n\\begin{eqnarray}
\n\\cos x &=& f(0) + f'(x)\\, x + \\frac{f”(0)}{2!}\\,x^2 + \\frac{f”'(0)}{3!}\\,x^3 +\\cdots \\\\
\n&=& \\cos 0 – \\sin 0 \\cdot x + \\frac{-\\cos 0}{2!}\\,x^2 + \\frac{\\sin 0}{3!}\\,x^3 + \\cdots \\\\
\n&=& 1 – \\frac{x^2}{2!} + \\frac{x^4}{4!} – \\frac{x^6}{6!} + \\cdots\\\\
\n&=& \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n)!}\\,x^{2n}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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series<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 7<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle 1 – \\frac{x^{2}}{2} + \\frac{x^{4}}{24} – \\frac{x^{6}}{720} + O\\left(x^{7}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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# \u968e\u4e57\u306e\u78ba\u8a8d<\/span>\r\n\r\nfactorial<\/span>(<\/span>2<\/span>),<\/span> factorial<\/span>(<\/span>4<\/span>),<\/span> factorial<\/span>(<\/span>6<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
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(2, 24, 720)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3057\u305f\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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$f(x) = e^x$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h5>\n
.removeO()<\/code> \u3067\u9ad8\u6b21\u306e\u9805\u3092\u53d6\u308a\u306e\u305e\u3044\u3066\u300c\u666e\u901a\u306e\u300d\u95a2\u6570\u306b\u3057\u3066\u304b\u3089 plot()<\/code> \u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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expt1<\/span> =<\/span> series<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 2<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nexpt2<\/span> =<\/span> series<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 3<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nexpt3<\/span> =<\/span> series<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 4<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nexpt1<\/span>,<\/span> expt2<\/span>,<\/span> expt3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
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(x + 1, x**2\/2 + x + 1, x**3\/6 + x**2\/2 + x + 1)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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plot<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> expt3<\/span>,<\/span> expt2<\/span>,<\/span> expt1<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>2<\/span>,<\/span> 2<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$f(x) = \\sin x$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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sint1<\/span> =<\/span> series<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 2<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nsint3<\/span> =<\/span> series<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 4<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nsint5<\/span> =<\/span> series<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 6<\/span>)<\/span>.<\/span>removeO<\/span>()<\/span>\r\nsint1<\/span>,<\/span> sint3<\/span>,<\/span> sint5<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
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(x, -x**3\/6 + x, x**5\/120 - x**3\/6 + x)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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plot<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> sint5<\/span>,<\/span> sint3<\/span>,<\/span> sint1<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>pi<\/span>,<\/span> pi<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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<\/p>\n
\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u6b21\u6570\u3092\u4e0a\u3052\u3066\u3044\u304f\u3068\uff0c$|x| < 1$ \u306e\u7bc4\u56f2\u3067\u306f\uff0c\u3082\u3068\u306e\u95a2\u6570\u306b\u8fd1\u3065\u3044\u3066\u3044\u304f\u3088\u3046\u3059\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4eba\u985e\u306e\u81f3\u5b9d\uff1a\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/h3>\n
\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\uff0c\u6307\u6570\u95a2\u6570 $e^{i x}$ \u3092 .rewrite(cos)<\/code> \u3092\u3064\u3051\u3066\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u304d\u76f4\u3057\u3066\u307f\u307e\u3059\u3002SymPy \u3067\u306f\u865a\u6570\u5358\u4f4d $i$ \u306f I<\/code> \u3067\u3059\u3002<\/p>\n
\n
from<\/span> sympy<\/span> import<\/span> pi<\/span>,<\/span> E<\/span>,<\/span> I<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[17]:<\/div>\n
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\n
Eq<\/span>(<\/span>exp<\/span>(<\/span>I<\/span>*<\/span>x<\/span>),<\/span> \r\n   exp<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>cos<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[17]:<\/div>\n
$\\displaystyle e^{i x} = i \\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u9006\u306b\uff0c\u4e09\u89d2\u95a2\u6570\u3092\u6307\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8868\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[18]:<\/div>\n
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cos<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>exp<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
$\\displaystyle \\frac{e^{i x}}{2} + \\frac{e^{- i x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
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\n
\n
sin<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>exp<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[19]:<\/div>\n
$\\displaystyle – \\frac{i \\left(e^{i x} – e^{- i x}\\right)}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n\\cos x &=& \\frac{e^{ix} + e^{-ix}}{2} \\\\
\n\\sin x &=& \\frac{e^{ix} – e^{-ix}}{2 i}
\n\\end{eqnarray}<\/p>\n
\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
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\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f<\/h4>\n
$$e^{i\\pi} + 1 = 0$$<\/p>\n
\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f\u306e\u3069\u3053\u304c\u3059\u3054\u3044\u304b\u3068\u3044\u3046\u3068\uff0c<\/p>\n
    \n
  • \u53ef\u80fd\u306b\u304a\u3051\u308b\u5358\u4f4d\u5143\u30bc\u30ed $0$ \u3068\u4e57\u6cd5\u306b\u304a\u3051\u308b\u5358\u4f4d\u5143 $1$ \u3068\u3044\u3046\u6574\u6570\u306e\u3082\u3063\u3068\u3082\u57fa\u672c\u3068\u306a\u308b\u6570<\/li>\n
  • \u7121\u7406\u6570\u306e\u4ee3\u8868\u9078\u624b\uff0c\u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u3068\u5186\u5468\u7387 $\\pi$<\/li>\n
  • \u305d\u3057\u3066\u865a\u6570\u5358\u4f4d $i$<\/li>\n<\/ul>\n
    \u3068\u3044\u3046\uff0c\u3044\u305a\u308c\u540d\u3060\u305f\u308b\u5f79\u8005\u9054\u304c\uff0c\u52a0\u6cd5\uff0c\u4e57\u6cd5\u304a\u3088\u3073\u51aa\u4e57\u306b\u3088\u3063\u3066\u898b\u4e8b\u306b\u7d50\u3073\u4ed8\u3051\u3089\u308c\u3066\u3044\u308b\u3068\u3044\u3046\u3053\u3068\uff01<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[20]:<\/div>\n
    \n
    \n
    \n
    E<\/span>**<\/span>(<\/span>I<\/span> *<\/span> pi<\/span>)<\/span> +<\/span> 1<\/span> ==<\/span> 0<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[20]:<\/div>\n
    \n
    True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/h3>\n
    \u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306f\uff0c\u305f\u3060\u307e\u304e\u3089\u308f\u3057\u3044\u307b\u3069\u306b\u4f3c\u305f\u8868\u8a18\u306a\u3060\u3051\u3067\u306a\u304f\uff0c\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f\u5bc6\u63a5\u306a\u95a2\u4fc2<\/strong>\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[21]:<\/div>\n
    \n
    \n
    \n
    cosh<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[21]:<\/div>\n
    $\\displaystyle \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[22]:<\/div>\n
    \n
    \n
    \n
    sinh<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[22]:<\/div>\n
    $\\displaystyle i \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[23]:<\/div>\n
    \n
    \n
    \n
    cos<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[23]:<\/div>\n
    $\\displaystyle \\cosh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[24]:<\/div>\n
    \n
    \n
    \n
    sin<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[24]:<\/div>\n
    $\\displaystyle i \\sinh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    … \u3068\u3044\u3046\u3053\u3068\u3067\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u305f\u3002<\/p>\n
    \\begin{eqnarray}
    \n\\cosh i x &=& \\cos x \\\\
    \n\\sinh i x &=& i \\sin x \\\\
    \n\\cos i x &=& \\cosh x \\\\
    \n\\sin i x &=& i \\sinh x
    \n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/h3>\n
    \u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3082\u307e\u305f\uff0c\u305f\u3060\u307e\u304e\u3089\u308f\u3057\u3044\u307b\u3069\u306b\u4f3c\u305f\u8868\u8a18\u306a\u3060\u3051\u3067\u306a\u304f\uff0c\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f\u5bc6\u63a5\u306a\u95a2\u4fc2<\/strong>\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[25]:<\/div>\n
    \n
    \n
    \n
    asin<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[25]:<\/div>\n
    $\\displaystyle i \\operatorname{asinh}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[26]:<\/div>\n
    \n
    \n
    \n
    atan<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[26]:<\/div>\n
    $\\displaystyle i \\operatorname{atanh}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[27]:<\/div>\n
    \n
    \n
    \n
    acos<\/span>(<\/span>I<\/span>*<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[27]:<\/div>\n
    $\\displaystyle – i \\operatorname{asinh}{\\left(x \\right)} + \\frac{\\pi}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    … \u3068\u3044\u3046\u3053\u3068\u3067\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u305f\u3002<\/p>\n
    \\begin{eqnarray}
    \n\\sin^{-1} i x &=& i \\sinh^{-1} x \\\\
    \n\\tan^{-1} i x &=& i \\tanh^{-1} x \\\\
    \n\\cos^{-1} i x &=& – i \\sinh^{-1} x + \\frac{\\pi}{2}
    \n\\end{eqnarray}<\/p>\n
    \u6700\u5f8c\u306e $\\displaystyle \\frac{\\pi}{2}$ \u304c\u4f55\u304b\u3057\u3063\u304f\u308a\u3053\u306a\u3044\u304c\uff0c\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304b\u3089\u7406\u89e3\u3067\u304d\u308b\u3002<\/p>\n
    $$\\cos^{-1} x + \\sin^{-1} x = \\frac{\\pi}{2}$$<\/p>\n
    SymPy \u3067\u3082\u78ba\u8a8d\u3057\u3066\u304a\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[28]:<\/div>\n
    \n
    \n
    \n
    Eq<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span> +<\/span> asin<\/span>(<\/span>x<\/span>),<\/span> \r\n   acos<\/span>(<\/span>x<\/span>)<\/span> +<\/span> asin<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>acos<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[28]:<\/div>\n
    $\\displaystyle \\operatorname{acos}{\\left(x \\right)} + \\operatorname{asin}{\\left(x \\right)} = \\frac{\\pi}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6034,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6063","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6063","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6063"}],"version-history":[{"count":2,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6063\/revisions"}],"predecessor-version":[{"id":8089,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6063\/revisions\/8089"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6034"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6063"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}