1 $B$O$8$a$K(B (PDF ¥Õ¥¡¥¤¥ë: taylor2.pdf)

2 $B

$B$^$:!"NI$/CN$i$l$F$$$k$h$&$K(B $x=0$ $B$N6a$/$G$O(B

$$\cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^6)$$ (1)

$B$G$"$k(B ($O(x^n)$ $B$O(B $n$ $Be$N9`$rI=$9(B) $B$N$G!"(B

$$\cos(at) = 1-\frac{a^2}{2}t^2+\frac{a^4}{24}t^4+O(t^6)$$ (2)

$B$H$J$k(B ( $t\rightarrow 0$ $B$N$H$-(B $at\rightarrow 0$ $B$J$N$G(B (1) $B$K$=$N$^$^(B $x=at$ $B$rBeF~$7$F$h$$(B)$B!#(B

$B$^$?!"(B

$$\mathop{\rm sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x+e^{-x}}$$

$B$G$"$j!"NI$/CN$i$l$F$$$k$h$&$K(B

$$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + O(x^6)$$

$B$J$N$G(B

$$\begin{eqnarray*} e^x + e^{-x} & = & 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac... ...& = & 2\left(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)\right) \end{eqnarray*}$$

$B$H$J$j!"$h$C$F(B

$$\mathop{\rm sech}(x) = \frac{1}{\displaystyle 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)}$$

$B$H$J$k!#(B

$B$3$3$G!"NI$/CN$i$l$F$$$k$h$&$K(B

$$\frac{1}{1+x} = 1-x+x^2+O(x^3)$$ (3)

$B$G$"$j!"(B $x\rightarrow 0$ $B$N$H$-(B

$$\frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)\rightarrow 0$$

$B$J$N$G!"$3$N(B (3) $B$N(B $x$ $B$NBe$o$j$K$3$N9`$rBeF~$9$k$3$H$,=PMh$F!"(B

$$\begin{eqnarray*} \mathop{\rm sech}(x) & = & \frac{1}{\displaystyle 1 + \frac{x... ...t(\left(\frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)\right)^3\right) \end{eqnarray*}$$

$B$H$J$k!#$3$3$G!"(B3 $BHVL\$N9`$OE83+$9$l$PJ,$+$k$,(B

$$\left(\frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)\right)^2 = \left(\frac{x^2}{2}\right)^2 + O(x^6) = \frac{x^4}{4} + O(x^6)$$

$B$G$"$j!"(B4 $BHVL\$N9`$bL@$i$+$K(B

$$O\left(\left(\frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)\right)^3\right) = O(x^6)$$

$B$H$J$k$N$G!"7k6I(B

$$\mathop{\rm sech}(x) = 1-\frac{x^2}{2} + \left(-\frac{1}{24}... ...ight)x^4 + O(x^6) = 1-\frac{x^2}{2} + \frac{5}{24}x^4 + O(x^6)$$

$B$G$"$k$3$H$,$o$+$j!"$h$C$F(B

$$\mathop{\rm sech}(bt) = 1-\frac{b^2}{2}t^2 + \frac{5b^4}{24}t^4 + O(t^6)$$ (4)

$B$H$J$k!#(B

$B8N$K!"(B(2), (4) $B$K$h$j(B

$$\begin{eqnarray*} f(t) & = & \cos(at)\mathop{\rm sech}(bt)\\ & = & \left(1-\f... ...rac{a^4}{24}+\frac{a^2b^2}{4}+\frac{5b^4}{24}\right)t^4 + O(t^6) \end{eqnarray*}$$

$B$H$J$k!#(B

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1 $B$O$8$a$K(B