## 2017年4月21日金曜日

### 数学 - 解析学 - 各種の初等関数 - 対数関数・指数関数(エルミートの多項式(Hermite polynomials))

1. $\begin{array}{l}n=1\\ \frac{d}{dx}\left({e}^{-{x}^{2}}\right)={e}^{-{x}^{2}}·-2x=-2{e}^{-{x}^{2}}x\\ -2{e}^{-{x}^{2}}x=-{H}_{n}\left(x\right){e}^{-{x}^{2}}\\ {H}_{n}\left(x\right)=2x\\ 1次の多項式。\\ {H}_{n}\left(-x\right)=-2x=-{H}_{n}\left(x\right)\\ 奇関数。\\ \\ \frac{{d}^{n+1}}{d{x}^{n+1}}\left({e}^{-{x}^{2}}\right)={\left(-1\right)}^{n}\left({H}_{n}\text{'}\left(x\right){e}^{-{x}^{2}}+{H}_{n}\left(x\right)·{e}^{-{x}^{2}}·\left(-2x\right)\right)\\ ={\left(-1\right)}^{n+1}\left(2{H}_{n}\left(x\right)x-{H}_{n}\text{'}\left(x\right)\right){e}^{-{x}^{2}}\\ {H}_{n+1}\left(x\right)=2{H}_{n}\left(x\right)x-{H}_{n}\text{'}\left(x\right)\\ n+1次の多項式。\\ {H}_{n+1}\left(-x\right)=2{H}_{n}\left(-x\right)\left(-x\right)-{H}_{n}\text{'}\left(-x\right)\\ =-2x{H}_{n}\left(-x\right)-{H}_{n}\text{'}\left(-x\right)\\ {H}_{n}\left(x\right)が奇関数のとき。\\ {H}_{n}\left(-x\right)=-{H}_{n}\left(x\right)\\ -{H}_{n}\text{'}\left(-x\right)=-{H}_{n}\text{'}\left(x\right)\\ {H}_{n}\text{'}\left(-x\right)={H}_{n}\text{'}\left(x\right)\\ {H}_{n+1}\left(-x\right)=2x{H}_{n}\left(x\right)-{H}_{n}\text{'}\left(x\right)={H}_{n+1}\left(x\right)\\ {H}_{n+1}は偶関数。\\ {H}_{n}\left(x\right)が偶関数のとき。\\ {H}_{n}\left(-x\right)={H}_{n}\left(x\right)\\ -{H}_{n}\text{'}\left(-x\right)={H}_{n}\text{'}\left(x\right)\\ {H}_{n}\text{'}\left(-x\right)=-{H}_{n}\text{'}\left(x\right)\\ {H}_{n+1}\left(-x\right)=-2x{H}_{n}\left(x\right)+{H}_{n}\text{'}\left(x\right)=-{H}_{n+1}\left(x\right)\\ {H}_{n+1}は奇関数。\end{array}$