{"id":2190,"date":"2023-04-09T11:29:11","date_gmt":"2023-04-09T02:29:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2190"},"modified":"2023-04-09T11:29:46","modified_gmt":"2023-04-09T02:29:46","slug":"maxima-%e3%81%a6%e3%82%99%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\/maxima-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/maxima-%e3%81%a6%e3%82%99%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":"Maxima \u3066\u3099\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u30fb\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f"},"content":{"rendered":"

<\/p>\n

\n
<\/div>\n
\n
\n

\u9ad8\u968e\u5c0e\u95a2\u6570<\/h3>\n

$f(x)$ \u3092 $x$ \u3067 $n$ \u968e\u5fae\u5206\u3059\u308b\u4f8b\u3002Maxima \u3067\u306f
\ndiff(f(x), x, n);<\/code> \u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002
\n\u4f8b\uff1a<\/p>\n

$$\\frac{d^2}{dx^2} \\sin x = \\dots$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
\/* \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>, 2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}-\\sin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
\/* \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\n'<\/span>diff<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 2<\/span>)<\/span> =<\/span> \r\n diff<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\frac{d^2}{d\\,x^2}\\,\\sin x=-\\sin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h3>\n
\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f taylor()<\/code> \u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
taylor<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 5<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\frac{x^5}{120}+\\frac{x^4}{24}+\\frac{x^3}{6}+\\frac{x^2}{2}+x+1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4e0a\u8a18\u306e\u7d50\u679c\u306f\uff0c\u901a\u5e38\u306a\u3089<\/p>\n
$$ 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\frac{x^5}{5!} +\\cdots$$<\/p>\n
\u3068\u306a\u308b\u3068\u3053\u308d\u3060\u304c\uff0c\u9805\u306e\u9806\u5e8f\u304c\u964d\u3079\u304d\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u3053\u306e\u9806\u5e8f\u306f Maxima-Jupyter \u4e0a\u3067\u306e\u307f\u306e\u3088\u3046\u3067\uff0c\u8a66\u3057\u306b\u30bf\u30fc\u30df\u30ca\u30eb\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6607\u3079\u304d\u306b\u4e26\u3073\u307e\u3059\u3002<\/p>\n
(%i1) taylor(exp(x), x, 0, 5);\r\n                               2    3    4    5\r\n                              x    x    x    x\r\n(%o1)\/T\/              1 + x + -- + -- + -- + --- + . . .\r\n                              2    6    24   120<\/code><\/pre>\n
Maxima-Jupyter \u3067\uff0c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u6607\u3079\u304d\u3067\u8868\u793a\u3055\u305b\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b trunc()<\/code> \u3092\u4f7f\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n
\uff08powerdisp: true$<\/code> \u3067\u3082\u53ef\u80fd\u3067\u3059\u304c\uff0c\u3053\u308c\u3060\u3068\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u4ee5\u5916\u306e\u5168\u3066\u306e\u591a\u9805\u5f0f\u304c\u6607\u3079\u304d\u3067\u8868\u793a\u3055\u308c\u3066\u3057\u307e\u3044\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
trunc<\/span>(<\/span>taylor<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 5<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}1+x+\\frac{x^2}{2}+\\frac{x^3}{6}+\\frac{x^4}{24}+\\frac{x^5}{120}+\\cdots \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u306f $e^x \\simeq \\displaystyle \\sum_{k=0}^n \\frac{x^k}{k!}$ \u306e $n=5$ \u306b\u76f8\u5f53\u3059\u308b\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\u3002Maxima \u3067\u306f\u968e\u4e57\u306f k!<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
\/* \u624b\u3063\u53d6\u308a\u65e9\u304f\u7dcf\u548c\u306e\u7b54\u3048\u3092\u77e5\u308a\u305f\u3044\u3068\u304d\u306f... *\/<\/span>\r\n\r\nsum<\/span>(<\/span>1\/<\/span>k<\/span>!<\/span>, k<\/span>, 0<\/span>,  10<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\frac{9864101}{3628800}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
\/* \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\n'<\/span>sum<\/span>(<\/span>1\/<\/span>k<\/span>!<\/span>, k<\/span>, 0<\/span>,  10<\/span>)<\/span> =<\/span>\r\n sum<\/span>(<\/span>1\/<\/span>k<\/span>!<\/span>, k<\/span>, 0<\/span>,  10<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\sum_{k=0}^{10}{\\frac{1}{k!}}=\\frac{9864101}{3628800}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
\/* \u6d6e\u52d5\u5c0f\u6570\u70b9\u6570\u3067\u8868\u793a *\/<\/span>\r\n\/* float() \u306f \u5c0f\u6570\u70b9\u4ee5\u4e0b 15 \u6841\u307e\u3067\u306e\u6d6e\u52d5\u5c0f\u6570\u70b9\u8868\u793a\u3002*\/<\/span>\r\n\r\nfloat<\/span>(<\/span>sum<\/span>(<\/span>1\/<\/span>k<\/span>!<\/span>, k<\/span>, 0<\/span>,  10<\/span>))<\/span>; \r\nfloat<\/span>(<\/span>%e<\/span>)<\/span>; \/* \u30cd\u30a4\u30d4\u30a2\u6570\u306e\u5024\u3002*\/<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}2.718281801146385\\]<\/div>\n<\/div>\n
\n
Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}2.718281828459045\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[8]:<\/div>\n
\n
\n
\n
\/* sin(x) \u306e x = 0 \u306e\u307e\u308f\u308a\u306e7\u6b21\u307e\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3002*\/<\/span>\r\n\r\ntrunc<\/span>(<\/span>taylor<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 7<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}x-\\frac{x^3}{6}+\\frac{x^5}{120}-\\frac{x^7}{5040}+\\cdots \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[9]:<\/div>\n
\n
\n
\n
\/* \u968e\u4e57\u306e\u78ba\u8a8d *\/<\/span>\r\n\r\n3!<\/span>;\r\n5!<\/span>;\r\n7!<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}6\\]<\/div>\n<\/div>\n
\n
Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}120\\]<\/div>\n<\/div>\n
\n
Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}5040\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$f(x) = \\cos x$ \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
\n
\n
In\u00a0[10]:<\/div>\n
\n
\n
\n
\/* cos(x) \u306e x = 0 \u306e\u307e\u308f\u308a\u306e7\u6b21\u307e\u3067\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3002*\/<\/span>\r\n\r\ntrunc<\/span>(<\/span>taylor<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 7<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}1-\\frac{x^2}{2}+\\frac{x^4}{24}-\\frac{x^6}{720}+\\cdots \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[11]:<\/div>\n
\n
\n
\n
\/* \u968e\u4e57\u306e\u78ba\u8a8d *\/<\/span>\r\n\r\n2!<\/span>;\r\n4!<\/span>;\r\n6!<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}2\\]<\/div>\n<\/div>\n
\n
Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}24\\]<\/div>\n<\/div>\n
\n
Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}720\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
<\/div>\n
\n
\n
$f(x) = e^x$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[12]:<\/div>\n
\n
\n
\n
expt1<\/span> :<\/span> taylor<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\nexpt2<\/span> :<\/span> taylor<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 2<\/span>)<\/span>;\r\nexpt3<\/span> :<\/span> taylor<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}x+1\\]<\/div>\n<\/div>\n
\n
Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}\\frac{x^2}{2}+x+1\\]<\/div>\n<\/div>\n
\n
Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}\\frac{x^3}{6}+\\frac{x^2}{2}+x+1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[13]:<\/div>\n
\n
\n
\n
plot2d<\/span>([<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, expt3<\/span>, expt2<\/span>, expt1<\/span>]<\/span>, [<\/span>x<\/span>, -<\/span>2<\/span>, 2]<\/span>, \r\n       [<\/span>legend<\/span>, \"exp(x)\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc3\u6b21\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc2\u6b21\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc1\u6b21\"<\/span>\r\n                ]<\/span>, \r\n       [<\/span>gnuplot_preamble<\/span>, \"set key top left;\"<\/span>]<\/span>\r\n      )<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
\n
<\/p>\n
\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$f(x) = \\sin x$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[14]:<\/div>\n
\n
\n
\n
sint1<\/span>:<\/span> taylor<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\nsint3<\/span>:<\/span> taylor<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 3<\/span>)<\/span>;\r\nsint5<\/span>:<\/span> taylor<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 5<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{22}$}x\\]<\/div>\n<\/div>\n
\n
Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}x-\\frac{x^3}{6}\\]<\/div>\n<\/div>\n
\n
Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{24}$}\\frac{x^5}{120}-\\frac{x^3}{6}+x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[15]:<\/div>\n
\n
\n
\n
plot2d<\/span>([<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, sint1<\/span>, sint3<\/span>, sint5<\/span>]<\/span>, [<\/span>x<\/span>, -<\/span>%pi<\/span>, %pi<\/span>]<\/span>, \r\n       [<\/span>legend<\/span>, \"sin x\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc1\u6b21\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc3\u6b21\"<\/span>, \r\n                \"\u30c6\u30a4\u30e9\u30fc5\u6b21\"<\/span>]<\/span>, \r\n       [<\/span>gnuplot_preamble<\/span>, \"set key top left;\"<\/span>]<\/span>\r\n      )<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
\n
<\/p>\n
\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u6b21\u6570\u3092\u4e0a\u3052\u3066\u3044\u304f\u3068\uff0c$|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\u4e09\u89d2\u95a2\u6570\u3092\u66f8\u304d\u76f4\u3057\u3066\u307f\u307e\u3059\u3002demoivre()<\/code> \u95a2\u6570\u306f\uff0c\uff08\u8907\u7d20\uff09\u6307\u6570\u95a2\u6570\u3092\u4e09\u89d2\u95a2\u6570\u306b\u5909\u63db\u3057\u307e\u3059\u3002<\/p>\n
Maxima \u3067\u306f\u865a\u6570\u5358\u4f4d $i$ \u306f %i<\/code> \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[16]:<\/div>\n
\n
\n
\n
exp<\/span>(<\/span>%i<\/span> *<\/span> theta<\/span>)<\/span> =<\/span> demoivre<\/span>(<\/span>exp<\/span>(<\/span>%i<\/span> *<\/span> theta<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}e^{i\\,\\vartheta}=i\\,\\sin \\vartheta+\\cos \\vartheta\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
exponentialize()<\/code> \u95a2\u6570\u306f\uff0cdemoivre()<\/code> \u95a2\u6570\u3068\u306f\u9006\u306b\uff0c\u4e09\u89d2\u95a2\u6570\uff08\u3084\u53cc\u66f2\u7dda\u95a2\u6570\uff09\u3092\u6307\u6570\u95a2\u6570\u3067\u8868\u793a\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[17]:<\/div>\n
\n
\n
\n
cos<\/span>(<\/span>theta<\/span>)<\/span> =<\/span> exponentialize<\/span>(<\/span>cos<\/span>(<\/span>theta<\/span>))<\/span>;\r\nsin<\/span>(<\/span>theta<\/span>)<\/span> =<\/span> exponentialize<\/span>(<\/span>sin<\/span>(<\/span>theta<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\cos \\vartheta=\\frac{e^{i\\,\\vartheta}+e^ {- i\\,\\vartheta }}{2}\\]<\/div>\n<\/div>\n
\n
Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{29}$}\\sin \\vartheta=-\\frac{i\\,\\left(e^{i\\,\\vartheta}-e^ {- i\\,\\vartheta }\\right)}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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[18]:<\/div>\n
    \n
    \n
    \n
    %e<\/span>**<\/span>(<\/span>%i<\/span> *<\/span> %pi<\/span>)<\/span> +<\/span> 1<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[18]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{30}$}0\\]<\/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<\/h4>\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[19]:<\/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[19]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{31}$}\\cos x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[20]:<\/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[20]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{32}$}i\\,\\sin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[21]:<\/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[21]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{33}$}\\cosh x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[22]:<\/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[22]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{34}$}i\\,\\sinh x\\]<\/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<\/h4>\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[23]:<\/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[23]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{35}$}i\\,{\\rm asinh}\\; x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[24]:<\/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[24]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{36}$}i\\,{\\rm atanh}\\; x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[25]:<\/div>\n
    \n
    \n
    \n
    asinh<\/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
    \\[\\tag{${\\it \\%o}_{37}$}i\\,\\arcsin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[26]:<\/div>\n
    \n
    \n
    \n
    atanh<\/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
    \\[\\tag{${\\it \\%o}_{38}$}i\\,\\arctan x\\]<\/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\\sinh^{-1} i x &=& i \\sin^{-1} x \\\\
    \n\\tanh^{-1} i x &=& i \\tan^{-1} x
    \n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u305f\u3060\u3057\uff0c$\\cos^{-1} i x$ \u3068 $\\cosh^{-1} i x$ \u306b\u3064\u3044\u3066\u306f\uff0cMaxima \u306f\u305d\u306e\u307e\u307e\u306e\u5f0f\u3092\u8fd4\u3057\uff0c\u7279\u306b\u66f8\u304d\u76f4\u3057\u3066\u304f\u308c\u306a\u3044\u3002<\/p>\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
    \\[\\tag{${\\it \\%o}_{39}$}\\arccos \\left(i\\,x\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[28]:<\/div>\n
    \n
    \n
    \n
    acosh<\/span>(<\/span>%i<\/span>*<\/span> x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[28]:<\/div>\n
    \\[\\tag{${\\it \\%o}_{40}$}{\\rm acosh}\\; \\left(i\\,x\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6032,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2190","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\/2190","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=2190"}],"version-history":[{"count":6,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2190\/revisions"}],"predecessor-version":[{"id":6097,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2190\/revisions\/6097"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6032"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}