2重根号
\(0\leq a\pm|b|\sqrt{c}\)のとき、
\[ \sqrt{a\pm|b|\sqrt{c}}=\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \]
\[ \sqrt{a\pm|b|\sqrt{c}}=\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \]
\(a^{2}-b^{2}c\)が平方数のとき2重根号が外せる
\[
\alpha_{\pm}=\sqrt{a\pm|b|\sqrt{c}}
\]
とおくと、
\begin{align*} \alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2} & =a\pm|b|\sqrt{c}+a\mp|b|\sqrt{c}\\ & =2a \end{align*} \begin{align*} \alpha_{\pm}\alpha_{\mp} & =\sqrt{\left(a\pm|b|\sqrt{c}\right)\left(a\mp|b|\sqrt{c}\right)}\\ & =\sqrt{a^{2}-b^{2}c} \end{align*} より、
\begin{align*} \alpha_{\pm} & =\frac{\left(\alpha_{\pm}+\alpha_{\mp}\right)+\left(\alpha_{\pm}-\alpha_{\mp}\right)}{2}\\ & =\frac{\left|\alpha_{\pm}+\alpha_{\mp}\right|\pm\left|\alpha_{\pm}-\alpha_{\mp}\right|}{2}\\ & =\frac{\sqrt{\left(\alpha_{\pm}+\alpha_{\mp}\right)^{2}}\pm\sqrt{\left(\alpha_{\pm}-\alpha_{\mp}\right)^{2}}}{2}\\ & =\frac{\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}+2\alpha_{\pm}\alpha_{\mp}}\pm\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}-2\alpha_{\pm}\alpha_{\mp}}}{2}\\ & =\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \end{align*}
\begin{align*} \alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2} & =a\pm|b|\sqrt{c}+a\mp|b|\sqrt{c}\\ & =2a \end{align*} \begin{align*} \alpha_{\pm}\alpha_{\mp} & =\sqrt{\left(a\pm|b|\sqrt{c}\right)\left(a\mp|b|\sqrt{c}\right)}\\ & =\sqrt{a^{2}-b^{2}c} \end{align*} より、
\begin{align*} \alpha_{\pm} & =\frac{\left(\alpha_{\pm}+\alpha_{\mp}\right)+\left(\alpha_{\pm}-\alpha_{\mp}\right)}{2}\\ & =\frac{\left|\alpha_{\pm}+\alpha_{\mp}\right|\pm\left|\alpha_{\pm}-\alpha_{\mp}\right|}{2}\\ & =\frac{\sqrt{\left(\alpha_{\pm}+\alpha_{\mp}\right)^{2}}\pm\sqrt{\left(\alpha_{\pm}-\alpha_{\mp}\right)^{2}}}{2}\\ & =\frac{\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}+2\alpha_{\pm}\alpha_{\mp}}\pm\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}-2\alpha_{\pm}\alpha_{\mp}}}{2}\\ & =\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \end{align*}
ページ情報
| タイトル | 2重根号 |
| URL | https://www.nomuramath.com/dv97ov6g/ |
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logの2乗の級数表示
\[
\log^{2}(1-x)=2\sum_{k=1}^{\infty}\frac{H_{k}}{k+1}x^{k+1}
\]
ウォリス積分の同表示
\[
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\int_{0}^{\frac{\pi}{2}}\cos^{n}\theta d\theta
\]
ベッセル関数のポアソン積分表示
\[
J_{\nu}(z)=\frac{1}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)}\left(\frac{z}{2}\right)^{\nu}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{izt}dt
\]
ウォリスの公式
\[
\prod_{k=1}^{\infty}\left(\frac{(2k)^{2}}{(2k-1)(2k+1)}\right)=\frac{\pi}{2}
\]