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\mondai
{f9311196 $B?e8M(B $BOB9,(B}
{97/01/18}
{f11196@ced.cas.uec.ac.jp}
{5-2}
{
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$B%G%k%?4X?t$K4X$7$F<!$N<0$r<($;(B
}
{
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$(a)\hspace{5mm}\delta(ax-b)=\frac{1}{a}\delta\left(x-\frac{b}{a}\right)$
\[\begin{array}{lcl}
\int_{-\infty}^{\infty}f(x)\delta(ax-b)dx 
& = & \frac{1}{a}\int_{-\infty}^{\infty}f(x/a)\delta(x-b)dx \\
& = & \frac{1}{a}\int_{-\infty}^{\infty}f(b/a)\delta(x)dx \\
& = & \int_{-\infty}^{\infty}f(x)\frac{1}{a}\delta\left(x-\frac{b}{a}\right) 
\end{array} \]
$(b)\hspace{5mm}\delta(x^{2}-a^{2})
=\frac{1}{2a}\left\{\delta(x-a)+\delta(x+a)\right\}$
\[\begin{array}{lcl}
\int_{-\infty}^{\infty}f(x)\delta(x^{2}-a^{2})dx
& = & \int_{-\infty}^{0}f(x)\delta(x^{2}-a^{2})dx
+\int_{0}^{\infty}f(x)\delta(x^{2}-a^{2})dx \\
& = & \frac{1}{2a}\left\{\int_{-\infty}^{0}f(x)\delta(x+a)dx
+\int_{0}^{\infty}f(x)\delta(x-a)dx\right\} \\
& = & \frac{1}{2a}\left\{\int_{-\infty}^{0}f(-a)\delta(x)dx
+\int_{0}^{\infty}f(a)\delta(x)dx\right\} \\
& = & \frac{1}{2a}\left\{f(-a)+f(a)\right\} \\
& = & \frac{1}{2a}\left\{\int_{-\infty}^{\infty}f(x)\delta(x+a)dx
+\int_{-\infty}^{\infty}f(x)\delta(x-a)dx\right\}
\end{array} \]
$(c)\hspace{5mm}\int_{-\infty}^{\infty}x\delta(x)dx=0$ \\
$f(x)=x$$B$H9M$($k$H!"(B\\
\[
\int_{-\infty}^{\infty}f(x)\delta(x)dx=0
\]
$(d)\hspace{5mm}\delta(-x)=\delta(x)$ \\
$\int_{-\infty}^{\infty}f(x)\delta(x-x_{0})=f(x_{0})$
$B$rMQ$$$F9M$($k$H!"(B \\
\[\begin{array}{lcl}
\int_{-\infty}^{\infty}f(x)\delta(-x)dx
& = & \int_{-\infty}^{\infty}f(0)\delta(-x)dx \\
& = & f(0) \\
& = & \int_{-\infty}^{\infty}f(x)\delta(x)dx
\end{array} \]
$(e)\hspace{5mm}\int_{-\infty}^{\infty}f(x)\frac{d}{dx}\delta(x)dx
=-f'(x)$
\[\begin{array}{lcl}
\int_{-\infty}^{\infty}f(x)\frac{d}{dx}\delta(x)dx
& = & [f(x)\delta(x)]_{-\infty}^{\infty}
-\int_{-\infty}^{\infty}f'(x)\delta(x)dx \\
& = & f(x)\left\{\delta(\infty)-\delta(-\infty)\right\}
-\int_{-\infty}^{\infty}f'(x)\delta(x)dx \\
& = & 0-\int_{-\infty}^{\infty}f'(x)\delta(x)dx \\
& = & -f'(0)
\end{array} \]
}

