巾関数の積分表現
巾関数の積分表現
\[ \frac{1}{z^{\alpha}}=\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{\infty}t^{\alpha-1}e^{-zt}dt \]
\[ \frac{1}{z^{\alpha}}=\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{\infty}t^{\alpha-1}e^{-zt}dt \]
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\(\Gamma\left(z\right)\)はガンマ関数\begin{align*}
\frac{1}{z^{\alpha}} & =\frac{1}{\Gamma\left(\alpha\right)}\frac{\Gamma\left(\alpha\right)}{z^{\alpha}}\\
& =\frac{1}{\Gamma\left(\alpha\right)}\mathcal{L}_{t}\left[H\left(t\right)t^{\alpha-1}\right]\left(z\right)\\
& =\frac{1}{\Gamma\left(\alpha\right)}\int_{-\infty}^{\infty}H\left(t\right)t^{\alpha-1}e^{-zt}dt\\
& =\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{\infty}t^{\alpha-1}e^{-zt}dt
\end{align*}
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\[
\frac{1}{2},\frac{3}{3},\frac{4}{3}
\]
有理数全体の集合
\[
f\left(x\right)=\frac{1}{\left\lfloor x\right\rfloor +1-\left\{ x\right\} }
\]
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\[
\sgn\left(x^{n+1}\right)\sum_{k=0}^{n}\frac{x^{k}}{k!}\leq\sgn\left(x^{n+1}\right)e^{x}
\]
逆2乗の別表示
\[
\frac{1}{\left(k+1\right)^{2}}=-\int_{0}^{1}x^{k}\log xdx
\]