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*}
ページ情報
| タイトル | 冪関数と指数関数の積の積分 |
| URL | https://www.nomuramath.com/nmp735fv/ |
| SNSボタン |
ライプニッツの法則
\[
\left(fg\right)^{(n)}=\sum_{k=0}^{n}C(n,k)f^{(k)}g^{(n-k)}
\]
偶関数の分母に指数関数+1がある対称な定積分
\[
\int_{-c}^{c}\frac{f_{e}\left(x\right)}{1+a^{x}}dx=\int_{0}^{c}f_{e}\left(x\right)dx
\]
微分・原始関数・定積分・不定積分の定義
\[
\frac{df(x)}{dx}=\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
\]
べき乗を含む0から∞までの定積分
\[
\Arg\left(\alpha\right)\ne\pi,0<b\Rightarrow\int_{0}^{\infty}f\left(x,\alpha x^{b}\right)dx=\frac{1}{\alpha^{\frac{1}{b}}b}\lim_{R\rightarrow\infty}\int_{0}^{Re^{i\Arg\left(\alpha\right)}}f\left(\frac{t^{\frac{1}{b}}}{\alpha^{\frac{1}{b}}},t\right)t^{\frac{1}{b}-1}dt
\]