x^4・exp(-α x^2) の広義積分[0,∞]
\(\int_0^{\infty} x^4 e^{-\alpha x^2} \mathrm{d}x\)
$$\begin{eqnarray} &&\int_0^{\infty} x^4 e^{-\alpha x^2} \mathrm{d}x \\&=&\int_0^{\infty} \left(\frac{t}{\alpha}\right)^2 e^{-t} \frac{1}{2 \sqrt{\alpha t}}\mathrm{d}t \\&&\;\cdots\;t=\alpha x^2,\;x=0 \rightarrow t=0,\;x=\infty \rightarrow t=\infty, \\&&\;\cdots\;x^2=\frac{t}{\alpha},\;x^4=x^{2^2}=\left(\frac{t}{\alpha}\right)^2 \\&&\;\cdots\;x=\sqrt{\frac{t}{\alpha}}\;(積分範囲からx\geq 0とする(?)) \\&&\;\cdots\;\frac{\mathrm{d}t}{\mathrm{d}x}=2\alpha x,\;\mathrm{d}x=\frac{1}{2\alpha x}\mathrm{d}t=\frac{1}{2\alpha \sqrt{\frac{t}{\alpha}}}\mathrm{d}t=\frac{1}{2 \sqrt{\alpha t}}\mathrm{d}t \\&=&\int_0^{\infty} \left(\frac{t}{\alpha}\right)^2 e^{-t} \frac{1}{2 \sqrt{\alpha t}}\mathrm{d}t \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\int_0^{\infty} \frac{t^2}{\sqrt{t}}e^{-t}\mathrm{d}t \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\int_0^{\infty} t^2 t^{-\frac{1}{2}}e^{-t}\mathrm{d}t \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\int_0^{\infty} t^{\frac{3}{2}}e^{-t}\mathrm{d}t \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\int_0^{\infty} t^{\frac{5}{2}-1}e^{-t}\mathrm{d}t \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\Gamma\left(\frac{5}{2}\right)\;\cdots\;\Gamma(z)=\int_0^{\infty} t^{z-1} e^{-t}\mathrm{d}t\;(\mathfrak{Re}(z)>0) \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\Gamma\left(\frac{1}{2}+2\right) \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\frac{(2\cdot 2-1)!!}{2^2}\sqrt{\pi}\;\cdots\;\Gamma\left(\frac{1}{2}+n\right)=\frac{(2n-1)!!}{2^n}\sqrt{\pi}\;(!!は二重階乗) \;(nは自然数) \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\frac{3!!}{4}\sqrt{\pi} \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\frac{3\cdot 1}{4}\sqrt{\pi}\;\cdots\;nの二重階乗は，1 から n まで n と同じ偶奇性を持つものだけを全て掛けた積 \\&=&\frac{1}{2\alpha^2\sqrt{\alpha}}\frac{3\sqrt{\pi}}{4} \\&=&\frac{3}{8\alpha^2}\sqrt{\frac{\pi}{\alpha}} \end{eqnarray}$$逆行列の補題 / matrix inversion lemma / Sherman–Morrison–Woodbury formula
逆行列の補題 / matrix inversion lemma / Sherman–Morrison–Woodbury formula
$$\begin{eqnarray} A&:&n\times n\;\textrm{matrix} \\C&:&k\times k\;\textrm{matrix} \\U&:&k\times n\;\textrm{matrix} \\V&:&n\times k\;\textrm{matrix} \end{eqnarray}$$ の時， $$\begin{eqnarray} \left(A+UCV\right)^{-1}&=&A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1} \end{eqnarray}$$\(\left(A+UCV\right)\left(右辺\right)=I_n\)
$$\begin{eqnarray} &&\left(A+UCV\right)\left(右辺\right) \\&=&\left(A+UCV\right)\left\{A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\} \\&=&\left(A+UCV\right)A^{-1}-\left(A+UCV\right)\left\{A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\} \\&=&AA^{-1}+UCVA^{-1} -A\left\{A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\} -UCV\left\{A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\} \\&=&I_n+\color{red}{U}\color{black}{C}\color{blue}{VA^{-1}}\color{black}{} -AA^{-1}\color{red}{U}\color{black}{}\left(C^{-1}+VA^{-1}U\right)^{-1}\color{blue}{VA^{-1}}\color{black}{} -UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1} \\&=&I_n +\color{red}{U}\color{green}{\left\{C-\left(C^{-1}+VA^{-1}U\right)^{-1}\right\}}\color{blue}{VA^{-1}}\color{black}{} -UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1} \\&=&I_n +U\color{green}{C\left\{\left(\cancel{C^{-1}}+VA^{-1}U\right)-\cancel{C^{-1}}\right\}\left(C^{-1}+VA^{-1}U\right)^{-1}}\color{black}{}VA^{-1} -UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1} \\&&\;\cdots\;X-Y^{-1}=X\color{orange}{YY^{-1}}\color{black}{}-\color{purple}{XX^{-1}}\color{black}{}Y^{-1}=X\left(Y-X^{-1}\right)Y^{-1} \\&=&I_n +U\color{green}{C\left(VA^{-1}U\right)\left(C^{-1}+VA^{-1}U\right)^{-1}}\color{black}{}VA^{-1} -UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1} \\&=&I_n +\cancel{UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}} -\cancel{UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}} \\&=&I_n \end{eqnarray}$$\(\left(右辺\right)\left(A+UCV\right)=I_n\)
$$\begin{eqnarray} &&\left(右辺\right)\left(A+UCV\right) \\&=&\left\{A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}\left(A+UCV\right) \\&=&A^{-1}A+A^{-1}UCV-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}A-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&=&I_n+A^{-1}UCV-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}V-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&=&I_n +\color{red}{A^{-1}U}\color{black}{C}\color{blue}{V}\color{black}{} -\color{red}{A^{-1}U}\color{black}{}\left(C^{-1}+VA^{-1}U\right)^{-1}\color{blue}{V}\color{black}{} -A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&=&I_n +\color{red}{A^{-1}U}\color{green}{ \left\{C-\left(C^{-1}+VA^{-1}U\right)^{-1}\right\} }\color{blue}{V}\color{black}{} -A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&=&I_n +A^{-1}U\color{green}{ \left(C^{-1}+VA^{-1}U\right)^{-1}\left\{\left(\cancel{C^{-1}}+VA^{-1}U\right)-\cancel{C^{-1}}\right\}C }\color{black}{}V -A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&&\;\cdots\;X-Y^{-1}=\color{orange}{Y^{-1}Y}\color{black}{}X-Y^{-1}\color{purple}{X^{-1}X}\color{black}{}=Y^{-1}\left(Y-X^{-1}\right)X \\&=&I_n +A^{-1}U\color{green}{ \left(C^{-1}+VA^{-1}U\right)^{-1}\left(VA^{-1}U\right)C }\color{black}{}V -A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV \\&=&I_n +\cancel{A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV} -\cancel{A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}UCV} \\&=&I_n \end{eqnarray}$$以上より
以上より右辺\(\left(A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right)\)は\(A+UCV\)の逆行列である．\(C=I_k\)の時
また\(C=I_k\)の時は，\(C^{-1}=I_k\)でもあるので， $$\begin{eqnarray} \left(A+UV\right)^{-1}&=&A^{-1}-A^{-1}U\left(I_k+VA^{-1}U\right)^{-1}VA^{-1} \end{eqnarray}$$ となる．
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