9 項別微分

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9 項別微分

例 4.38 (項別微分の具体例)

$\displaystyle e^{x}$ $\displaystyle = \sum_{n=0}^{\infty}\frac{1}{n!}x^{n}= 1+x+\frac{1}{2}x^{2}+\frac{1}{3!}x^3+ \frac{1}{4!}x^4+\cdots$ (692)

$\displaystyle \frac{d}{dx}e^{x}$ $\displaystyle = \frac{d}{dx} \left(1+x+\frac{1}{2}x^{2}+\frac{1}{3!}x^3+ \frac{1}{4!}x^4+\cdots+ \frac{1}{n!}x^{n}+ \frac{1}{(n+1)!}x^{n+1}+\cdots \right)$ (693)

$\displaystyle = \frac{d}{dx}1+ \frac{d}{dx}x+ \frac{d}{dx}\left(\frac{1}{2}x^{2... ...x}\left(\frac{1}{3!}x^3\right)+ \frac{d}{dx}\left(\frac{1}{4!}x^4\right)+\cdots$ (694)

$\displaystyle \qquad\qquad\qquad\cdots+ \frac{d}{dx}\left(\frac{1}{n!}x^{n}\right)+ \frac{d}{dx}\left(\frac{1}{(n+1)!}x^{n+1}\right)+\cdots$ (695)

$\displaystyle = 0+1+\frac{2}{2}x^{2}+ \frac{3}{3!}x^2+\frac{4}{4!}x^{3}+\cdots+ \frac{n}{n!}x^{n-1}+\frac{n+1}{(n+1)!}x^{n}+\cdots$ (696)

$\displaystyle = 1+x+ \frac{1}{2!}x^2+\frac{1}{3!}x^3+\cdots+ \frac{1}{(n-1)!}x^{n-1}+\frac{1}{n!}x^{n}+\cdots$ (697)

$\displaystyle = \sum_{n=0}^{\infty}\frac{1}{n!}x^{n}$ (698)

$\displaystyle =e^{x}$ (699)

例 4.39 (項別微分の具体例)

$\displaystyle \sin x$ $\displaystyle = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+ (-1)^{n}\frac{x^{2n+1}}{(2n+1)!}+\cdots$ (700)

$\displaystyle \frac{d}{dx}\sin x$ $\displaystyle = \frac{d}{dx}\left( x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+ (-1)^{n}\frac{x^{2n+1}}{(2n+1)!}+\cdots\right)$ (701)

$\displaystyle = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots+ (-1)^{n}\frac{x^{2n}}{(2n)!}+\cdots=\cos x$ (702)

例 4.40 (項別微分の具体例)

$\displaystyle \log(1+x)$ $\displaystyle = x-\frac{x^2}{2}+\cdots+(-1)^{n}\frac{x^{n}}{n}+\cdots$ (703)

$\displaystyle \frac{d}{dx} \log(1+x)$ $\displaystyle = 1-x+\cdots+(-1)^{n}x^{n-1}+\cdots=\frac{1}{1+x}$ (704)

問 4.41 これを示せ．

例 4.42 (テーラー級数の微分)

$\displaystyle f(x)$ $\displaystyle = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$ (705)

$\displaystyle f'(x)= \frac{d}{dx}f(x)$ $\displaystyle = \frac{d}{dx} \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$ (706)

$\displaystyle = \frac{d}{dx} \left( f(a)+\sum_{n=1}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n} \right)$ (707)

$\displaystyle =0+ \sum_{n=1}^{\infty} \frac{d}{dx}\left( \frac{f^{(n)}(a)}{n!}(x-a)^{n}\right)$ (708)

$\displaystyle = \sum_{n=1}^{\infty} f^{(n)}(a) \frac{n}{n!} (x-a)^{n-1}$ (709)

$\displaystyle = \sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{(n-1)!} (x-a)^{n-1}$ (710)

$\displaystyle \quad(m=n-1;n=1,2,3,\cdots)$ (711)

$\displaystyle = \sum_{m=0}^{\infty} \frac{f^{(m+1)}(a)}{m!} (x-a)^{m}$ (712)

$\displaystyle \quad(m\to n)$ (713)

$\displaystyle = \sum_{n=0}^{\infty} \frac{(f')^{(n)}(a)}{n!} (x-a)^{n}$ (714)

$\displaystyle =f'(x)$ (715)

に関するテーラー級数を微分したものは，に関するテーラー級数となる．テーラー級数は微分演算に対して不変である．

Next: 10 項別積分 Up: 4 テイラー級数 Previous: 8 解析関数 Contents
Kondo Koichi
Created at 2003/08/29