冪関数と指数関数の積の積分
冪関数と指数関数の積の積分
(1)
\[ \int z^{\alpha}e^{\beta z}dz=\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C \](2)
\[ \int z^{\alpha}\beta^{z}dz=\frac{z^{\alpha}}{\Log\beta\left(-z\Log\beta\right)^{\alpha}}\Gamma\left(\alpha+1,-z\Log\beta\right)+C \]-
\(\Gamma\left(x,y\right)\)は第2種不完全ガンマ関数(1)
\begin{align*} \int z^{\alpha}e^{\beta z}dz & =\frac{\left(-\beta\right)^{\alpha}z^{\alpha}}{\left(-\beta z\right)^{\alpha}}\int\frac{\left(-\beta z\right)^{\alpha}}{\left(-\beta\right)^{\alpha+1}}e^{\beta z}d\left(-\beta z\right)\\ & =\frac{z^{\alpha}}{\left(-\beta\right)\left(-\beta z\right)^{\alpha}}\int^{-\beta z}z^{\alpha}e^{-z}dz\\ & =\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C \end{align*}(2)
\begin{align*} \int z^{\alpha}\beta^{z}dz & =\int z^{\alpha}e^{z\Log\beta}dz\\ & =\left[\int z^{\alpha}e^{\beta z}dz\right]_{\beta\rightarrow\Log\beta}\\ & =\left[\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C\right]_{\beta\rightarrow\Log\beta}\\ & =\frac{z^{\alpha}}{\Log\beta\left(-z\Log\beta\right)^{\alpha}}\Gamma\left(\alpha+1,-z\Log\beta\right)+C \end{align*}
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べき乗を含む0から∞までの定積分
\[
\Arg\left(\alpha\right)\ne\pi,0<b\Rightarrow\int_{0}^{\infty}f\left(x,\alpha x^{b}\right)dx=\frac{1}{\alpha^{\frac{1}{b}}b}\lim_{R\rightarrow\infty}\int_{0}^{Re^{i\Arg\left(\alpha\right)}}f\left(\frac{t^{\frac{1}{b}}}{\alpha^{\frac{1}{b}}},t\right)t^{\frac{1}{b}-1}dt
\]
微分の基本公式
\[
\left(f(x)g(x)\right)'=f'(x)g(x)+f(x)g'(x)
\]
ルートの中に2乗を含む積分
\[
\int f\left(\sqrt{a^{2}-x^{2}}\right)dx=a\int f\left(a\cos t\right)\cos tdt\cnd{x=a\sin t}
\]
微分形接触型積分
\[
\int f'(g(x))g'(x)dx=f(g(x))
\]