<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2}dx=\\sqrt{\\frac{2\\pi}{a}}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}x^{2n}e^{-\\frac{1}{2}ax^2}dx=\\sqrt{2\\pi}a^{-\\frac{2n+1}{2}}(2n-1)!!\\)<\/p>\n\n\n\n
\\(\\int_{0}^{+\\infty}x^{2n+1}e^{-\\frac{1}{2}ax^2}dx=2^nn!a^{-n-1}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2+Jx}dx =\\sqrt{\\frac{2\\pi}{a}}e^{\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2+iJx}dx =\\sqrt{\\frac{2\\pi}{a}}e^{-\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{i(\\frac{1}{2}ax^2+Jx)}dx =\\sqrt{\\frac{2\\pi i}{a}}e^{-i\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx}d^nx=\\sqrt{\\frac{(2\\pi)^n}{detK}}\\)\uff08K\u662f\u5bf9\u79f0\u77e9\u9635\uff0c\u4ee5\u4e0b\u540c\u7406\uff09<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx+Jx}d^nx=\\sqrt{\\frac{(2\\pi)^n}{detK}}e^{\\frac{1}{2}JK^{-1}J}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx+iJx}d^nx=\\sqrt{\\frac{(2\\pi)^n}{detK}}e^{-\\frac{1}{2}JK^{-1}J}\\)<\/p>\n\n\n\n
\\(\\int_{-\\infty}^{+\\infty}e^{i(\\frac{1}{2}x^TKx+Jx)}d^nx=\\sqrt{\\frac{(2\\pi i)^n}{detK}}e^{-\\frac{i}{2}JK^{-1}J}\\)<\/p>\n\n\n\n
\u8bef\u5dee\u51fd\u6570\uff1a\\(erf(x)=\\frac{2}{\\sqrt{\\pi}}\\int_0^xexp(-y^2)dy\\)<\/p>\n\n\n\n
\u9ad8\u65af\u79ef\u5206\u53ef\u4ee5\u8bf4\u662f\u6700\u5e38\u89c1\u7684\u4e00\u7c7b\u79ef\u5206\u4e86\uff0c\u5728\u5149\u5b66\u3001\u7edf\u8ba1\u7269\u7406\u3001\u91cf\u5b50\u573a\u8bba\u7b49\u8bb8\u8bb8\u591a\u591a\u7684\u9886\u57df\u90fd\u6709\u7740\u91cd\u8981\u7684\u5730\u4f4d\uff0c\u6211\u66fe\u7ecf\u542c\u8fc7\u4e00\u79cd\u6709\u8da3\u7684\u8bf4\u6cd5\uff1a\u201c\u7269\u7406\u5b66\u5bb6\u53ea\u5b66\u4f1a\u4e86\u6c42\u89e3\u9ad8\u65af\u79ef\u5206\u201d\uff0c\u6240\u8a00\u975e\u865a\u3002\u9ad8\u65af\u79ef\u5206\u7684\u91cd\u8981\u6027\u4e0d\u4ec5\u5728\u4e8e\u5b83\u7684\u5e38\u89c1\uff0c\u5176\u4f18\u7f8e\u7684\u6027\u8d28\u4e5f\u4f7f\u5b83\u6210\u4e3a\u8fd9\u4e2a\u4e16\u754c\u5947\u5999\u6027\u7684\u6765\u6e90\uff0c\u8b6c\u5982\uff0c\u7531\u4e8e\u9ad8\u65af\u79ef\u5206\u7684\u4efb\u610f\u5947\u6570\u9636\u77e9\u90fd\u4e3a0\uff0c\u5076\u6570\u9636\u77e9\u53ef\u4ee5\u5199\u4e3a\u6240\u6709\u4e8c\u9636\u77e9\u7684\u4e58\u79ef\u3002\u672c\u7bc7\u6587\u7ae0\u5176\u5b9e\u662f\u5e26\u6709\u5de5\u5177\u6027\u8d28\u662f\u4e00\u7bc7\u6587\u7ae0\uff0c\u8bb0\u5f55\u4e86\u6211\u6240\u9047\u5230\u7684\u5404\u79cd\u9ad8\u65af\u79ef\u5206\u7684\u5f62\u5f0f\uff0c\u65b9\u4fbf\u968f\u53d6\u968f\u7528\uff1b\u5f53\u7136\uff0c\u6211\u4f1a\u9002\u5f53\u5730\u5199\u4e00\u4e9b\u6c42\u89e3\u8fc7\u7a0b\uff0c\u5e0c\u671b\u80fd\u591f\u7ed9\u5bf9\u8fd9\u4e2a\u9886\u57df\u8fd8\u4e0d\u592a\u719f\u6089\u7684\u521d\u5b66\u8005\u4e00\u4e9b\u53c2\u8003\u548c\u5e2e\u52a9\u3002<\/p>\n\n\n\n
\u770b\u8fd9\u79cd\u5f62\u5f0f\u7684\u9ad8\u65af\u79ef\u5206\uff1a \\(I=\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^2}dx\\)<\/p>\n\n\n\n
\u8ba1\u7b97\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u8981\u505a\u4e00\u4e2a\u5e73\u65b9\uff1a \\(I=\\sqrt{I^2}=\\sqrt{\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^2}dx\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}y^2}dy}=\\sqrt{\\int_{-\\infty}^{+\\infty}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}(x^2+y^2)}dxdy}\\)<\/p>\n\n\n\n
\u53d6 \\(x=rcos\\theta, y=rsin\\theta\\)<\/p>\n\n\n\n
\u5219\u6709 \\(x^2+y^2=r^2,dxdy=Jdrd\\theta\\)<\/p>\n\n\n\n
\u5176\u4e2d\u96c5\u53ef\u6bd4\u77e9\u9635 \\(J=\\left|\\frac{\\partial(x,y)}{\\partial(r,\\theta)}\\right|=\\left|\\begin{array}{cc} \\frac{\\partial x}{\\partial r} & \\frac{\\partial y}{\\partial r} \\ \\frac{\\partial x}{\\partial \\theta} & \\frac{\\partial y}{\\partial \\theta} \\end{array}\\right| =\\left|\\begin{array}{cc} cos\\theta & sin\\theta \\ -rsin\\theta & rcos\\theta \\end{array}\\right| =r\\)<\/p>\n\n\n\n
\u56e0\u6b64\u4e0a\u5f0f\u7b49\u4e8e\uff1a \\(I=\\sqrt{\\int_{0}^{+\\infty}\\int_{0}^{2\\pi}e^{-\\frac{1}{2}r^2}rdrd\\theta} =\\sqrt{2\\pi \\int_{0}^{+\\infty}re^{-\\frac{1}{2}r^2}dr} =\\sqrt{2\\pi \\int_{0}^{+\\infty}e^{-\\frac{1}{2}r^2}d(\\frac{1}{2}r^2)} =\\sqrt{2\\pi (-e^{-\\infty}+e^{-0})} =\\sqrt{2\\pi}\\)<\/p>\n\n\n\n
\u7a0d\u4f5c\u53d8\u5f62\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230 \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2}dx=\\frac{1}{\\sqrt{a}}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}(\\sqrt{a}x)^2}d(\\sqrt{a}x)=\\sqrt{\\frac{2\\pi}{a}}\\)<\/p>\n\n\n\n
\uff082)<\/strong> \u5c06\u4e0a\u5f0f\u4e24\u8fb9\u5206\u522b\u5bf9a\u6c42\u5bfc\uff0c\u5c31\u5f97\u5230 \\(\\int_{-\\infty}^{+\\infty}(-\\frac{1}{2}x^2)e^{-\\frac{1}{2}ax^2}dx =\\sqrt{2\\pi}(-\\frac{1}{2})a^{-\\frac{3}{2}}\\)<\/p>\n\n\n\n \u6c42n\u6b21\u5bfc\uff0c\u5c31\u5f97\u5230 \\(\\int_{-\\infty}^{+\\infty}x^{2n}e^{-\\frac{1}{2}ax^2}dx=\\sqrt{2\\pi}a^{-\\frac{2n+1}{2}}(2n-1)!!\\)<\/p>\n\n\n\n \uff083\uff09 <\/strong>\u5bb9\u6613\u5f97\u5230 \\(\\int_{0}^{+\\infty}xe^{-\\frac{1}{2}ax^2}dx=\\frac{1}{a}\\) (\u4eff\u7167\uff081\uff09\u4e2d\u5bf9r\u7684\u79ef\u5206)<\/p>\n\n\n\n \u4f9d\u7167\uff082\uff09\u7684\u601d\u8def\uff0c\u6c42n\u6b21\u5bfc\uff0c\u5c31\u5f97\u5230 \\(\\int_{0}^{+\\infty}x^{2n+1}e^{-\\frac{1}{2}ax^2}dx=2^nn!a^{-n-1}\\)<\/p>\n\n\n\n \uff084\u30015\u30016\uff09 \u6709\u6e90\u9ad8\u65af\u79ef\u5206<\/strong><\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2+Jx}dx =\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}a(x^2-\\frac{2J}{a}x+\\frac{J^2}{a^2})+\\frac{J^2}{2a}}dx =e^{\\frac{J^2}{2a}}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}a(x-\\frac{J}{a})^2}dx =e^{\\frac{J^2}{2a}}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}a(x-\\frac{J}{a})^2}d(x-\\frac{J}{a}) =\\sqrt{\\frac{2\\pi}{a}}e^{\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}ax^2+iJx}dx =\\sqrt{\\frac{2\\pi}{a}}e^{\\frac{(iJ)^2}{2a}} =\\sqrt{\\frac{2\\pi}{a}}e^{-\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{i(\\frac{1}{2}ax^2+Jx)}dx =\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}a(xe^{-\\frac{\\pi}{4}})^2+Je^{\\frac{\\pi}{2}}x)}dx =\\sqrt{\\frac{2\\pi i}{a}}e^{-i\\frac{J^2}{2a}}\\)<\/p>\n\n\n\n \uff087\u30018\u30019\u300110\uff09 \u9ad8\u65af\u91cd\u79ef\u5206<\/strong><\/p>\n\n\n\n \u9996\u5148\uff0c\u5728\uff081\uff09\u4e2d\u6211\u4eec\u5df2\u7ecf\u6c42\u8fc7\u4e8c\u91cd\u9ad8\u65af\u79ef\u5206\u4e86\uff1a<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}(x^2+y^2)}dxdy=I^2=(\\sqrt{2\\pi})^2=2\\pi\\)<\/p>\n\n\n\n n\u91cd\u9ad8\u65af\u79ef\u5206\u4e5f\u4e0d\u96be\u6c42\u5f97\uff1a<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}\\sum_\\limits{i=1}^\\limits{n} x_i^2}dx_1dx_2...dx_n=I^n=\\sqrt{(2\\pi)^n}\\)<\/p>\n\n\n\n \u53ef\u5982\u679c\u6307\u6570\u7684\u5206\u5b50\u4e0a\u662f\u4e00\u4e2a\u4efb\u610f\u7684\u4e8c\u6b21\u578b\uff0c\u5c31\u4e0d\u80fd\u5411\u4e0a\u9762\u4e00\u6837\u7b80\u5355\u5730\u505a\u51fa\u6765\u4e86\u3002\u8bbe\u4e8c\u6b21\u578b\u4e3a\u00a0\\(x^TKx\\)\u00a0\uff08\\(x=(x_1,x_2,...,x_n),K=(K_{ij}),i,j\\in(1,n)\\)\u00a0\uff09\uff0c\u82e5\\(K\\)\u662f\u5bf9\u79f0\u77e9\u9635\uff08\u00a0\\(K=K^T\\)\u00a0\uff09\uff0c\u5219\u53ef\u4ee5\u505a\u5206\u89e3\u00a0\\(K=S^TS\\)\u00a0\uff0c\u4f7f\u5f97\u00a0\\(x^TKx=x^TS^TSx=(Sx)^T(Sx)\\)\u00a0\u3002\u6211\u4eec\u5b9a\u4e49\u00a0\\(y=Sx\\)\u00a0\uff0c\u5219\u539fn\u91cd\u9ad8\u65af\u79ef\u5206\u5c31\u53d8\u4e3a\u4e86<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx}d^nx= \\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}y^Ty}Jd^ny\\)<\/p>\n\n\n\n \u5176\u4e2d\u00a0\\(J=\\left|\\frac{\\partial x}{\\partial y}\\right|=\\frac{1}{det S}=\\frac{1}{\\sqrt{detK}}\\)<\/p>\n\n\n\n \u56e0\u6b64\u6700\u7ec8\u7684\u7b54\u6848\u4e3a\u00a0\\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx}d^nx= \\frac{1}{\\sqrt{detK}}\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}y^Ty}d^ny=\\sqrt{\\frac{(2\\pi)^n}{detK}}\\)<\/p>\n\n\n\n \u540c\u6837\u5730\uff0c\u6709\u6e90\u9ad8\u65af\u91cd\u79ef\u5206\u4ec5\u9700\u628a\uff084\u30015\u30016\uff09\u4e2d\u7684\u00a0\\(a\\)\u00a0\u6362\u4e3a\u00a0\\(K\\)\u6216\\(det K\\)\u00a0\uff08\u53d6\u51b3\u4e8e\u6700\u540e\u7684\u7ed3\u679c\u9700\u8981\u662f\u6807\u91cf\uff09\u5c31\u53ef\u4ee5\u4e86\uff1a<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx+Jx}d^nx=\\sqrt{\\frac{(2\\pi)^n}{detK}}e^{\\frac{1}{2}JK^{-1}J}\\)<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{2}x^TKx+iJx}d^nx=\\sqrt{\\frac{(2\\pi)^n}{detK}}e^{-\\frac{1}{2}JK^{-1}J}\\)<\/p>\n\n\n\n \\(\\int_{-\\infty}^{+\\infty}e^{i(\\frac{1}{2}x^TKx+Jx)}d^nx=\\sqrt{\\frac{(2\\pi i)^n}{detK}}e^{-\\frac{i}{2}JK^{-1}J}\\)<\/p>\n\n\n\n \uff0811\uff09\u8bef\u5dee\u51fd\u6570<\/strong><\/p>\n\n\n\n \u4e0d\u53ef\u6c42\u89e3\u7684\u9ad8\u65af\u79ef\u5206\u53ef\u4ee5\u7528\u8bef\u5dee\u51fd\u6570\u6765\u8868\u793a\uff1a<\/p>\n\n\n\n \u8bef\u5dee\u51fd\u6570\uff1a\u00a0\\(erf(x)=\\frac{2}{\\sqrt{\\pi}}\\int_0^xexp(-y^2)dy\\)<\/p>\n\n\n\n \uff08\u672a\u5b8c\u5f85\u7eed\uff09<\/p>\n\n\n\n \u628a\u77e5\u4e4e\u6587\u7ae0\u94fe\u63a5\u8fde\u63a5\u8d34\u4e0a\u5427\uff1a<\/p>\n\n\n\n