(1)

デルタ関数の定義より明らかに成り立つ。

(2)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(x-a\right)dx & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{-\infty}^{a-\epsilon}f\left(x\right)\delta\left(x-a\right)dx+\int_{a-\epsilon}^{a+\epsilon}f\left(x\right)\delta\left(x-a\right)dx+\int_{a+\epsilon}^{\infty}f\left(x\right)\delta\left(x-a\right)dx\right\} \\ & =\lim_{\epsilon\rightarrow+0}\int_{a-\epsilon}^{a+\epsilon}f\left(x\right)\delta\left(x-a\right)dx \end{align*}

(3)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(x-a\right)dx & =f\left(a\right)\\ & =\int_{-\infty}^{\infty}f\left(a\right)\delta\left(x-a\right)dx \end{align*} となるので
\[ f\left(x\right)\delta\left(x-a\right)=f\left(a\right)\delta\left(x-a\right) \] となり与式は成り立つ。

(4)

デルタ関数の定義
\[ \int_{-\infty}^{\infty}f\left(x\right)\delta\left(x\right)dx=f\left(0\right) \] で\(f\left(x\right)=1\)とおけばよい。

(5)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(-x\right)dx & =-\int_{\infty}^{-\infty}f\left(-x\right)\delta\left(x\right)dx\cmt{x\rightarrow-x}\\ & =\int_{-\infty}^{\infty}f\left(-x\right)\delta\left(x\right)dx\\ & =\int_{-\infty}^{\infty}f\left(0\right)\delta\left(x\right)dx\\ & =\int_{-\infty}^{\infty}f\left(x\right)\delta\left(x\right)dx \end{align*} となるので
\[ \delta\left(-x\right)=\delta\left(x\right) \] となり与式は成り立つ。

(6)

\begin{align*} \int_{-\infty}^{\infty}x\delta\left(x\right)dx & =0\\ & =\int_{-\infty}^{\infty}0\delta\left(x\right)dx \end{align*} となるので
\[ x\delta\left(x\right)=0 \] となり与式は成り立つ。

(7)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(ax\right)dx & =\lim_{r\rightarrow\infty}\int_{-r}^{r}f\left(x\right)\delta\left(ax\right)dx\\ & =\lim_{r\rightarrow\infty}\int_{-\sgn\left(a\right)r}^{\sgn\left(a\right)r}f\left(\frac{y}{a}\right)\delta\left(y\right)\frac{1}{a}dy\cmt{ax=y}\\ & =\lim_{r\rightarrow\infty}\sgn\left(a\right)\int_{-r}^{r}f\left(\frac{y}{a}\right)\delta\left(y\right)\frac{1}{a}dy\\ & =\int_{-\infty}^{\infty}f\left(\frac{y}{a}\right)\delta\left(y\right)\frac{1}{\left|a\right|}dy\\ & =\int_{-\infty}^{\infty}f\left(0\right)\frac{1}{\left|a\right|}\delta\left(y\right)dy\\ & =\int_{-\infty}^{\infty}f\left(y\right)\frac{1}{\left|a\right|}\delta\left(y\right)dy \end{align*} となるので
\[ \delta\left(ax\right)=\frac{1}{\left|a\right|}\delta\left(x\right) \] となり与式は成り立つ。

(8)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(g\left(x\right)\right)dx & =\sum_{i}\int_{a_{i}-\epsilon}^{a_{i}+\epsilon}f\left(x\right)\delta\left(g\left(x_{i}\right)+g'\left(x_{i}\right)\left(x-x_{i}\right)+\cdots\right)dx\\ & =\sum_{i}\int_{a_{i}-\epsilon}^{a_{i}+\epsilon}f\left(x\right)\delta\left(g'\left(x_{i}\right)\left(x-x_{i}\right)\right)dx\\ & =\sum_{i}\int_{a_{i}-\epsilon}^{a_{i}+\epsilon}\frac{f\left(x\right)}{\left|g'\left(x_{i}\right)\right|}\delta\left(\left(x-x_{i}\right)\right)dx\\ & =\int_{-\infty}^{\infty}f\left(x\right)\sum_{i}\frac{1}{\left|g'\left(x_{i}\right)\right|}\delta\left(\left(x-x_{i}\right)\right)dx \end{align*} これより、
\[ \delta\left(g\left(x\right)\right)=\sum_{i}\frac{1}{\left|g'\left(x_{i}\right)\right|}\delta\left(\left(x-x_{i}\right)\right) \] となるので与式は成り立つ。

(9)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta\left(\left(x-a\right)\left(x-b\right)\right)dx & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{\min\left(a,b\right)-\epsilon}^{\min\left(a,b\right)+\epsilon}f\left(x\right)\delta\left(\left(x-a\right)\left(x-b\right)\right)dx+\int_{a-\epsilon}^{a+\epsilon}f\left(x\right)\delta\left(\left(x-a\right)\left(x-b\right)\right)dx\right\} \\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{\left|b-a\right|\epsilon}^{-\left|b-a\right|\epsilon}f\left(\frac{a+b-\sqrt{\left(a+b\right)^{2}-4\left(ab-y\right)}}{2}\right)\delta\left(y\right)\frac{-1}{\sqrt{\left(a+b\right)^{2}-4\left(ab-y\right)}}dy+\int_{-\left|b-a\right|\epsilon}^{\left|b-a\right|\epsilon}f\left(\frac{a+b+\sqrt{\left(a+b\right)^{2}-4\left(ab-y\right)}}{2}\right)\delta\left(y\right)\frac{1}{\sqrt{\left(a+b\right)^{2}-4\left(ab-y\right)}}dy\right\} \cmt{y=\left(x-a\right)\left(x-b\right)}\\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{-\left|b-a\right|\epsilon}^{\left|b-a\right|\epsilon}f\left(\frac{a+b-\sqrt{\left(a+b\right)^{2}-4ab}}{2}\right)\delta\left(y\right)\frac{1}{\sqrt{\left(a+b\right)^{2}-4ab}}dy+\int_{-\left|b-a\right|\epsilon}^{\left|b-a\right|\epsilon}f\left(\frac{a+b+\sqrt{\left(a+b\right)^{2}-4ab}}{2}\right)\delta\left(y\right)\frac{1}{\sqrt{\left(a+b\right)^{2}-4ab}}dy\right\} \\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{-\epsilon}^{\epsilon}f\left(\frac{a+b-\left|a-b\right|}{2}\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy+\int_{-\epsilon}^{\epsilon}f\left(\frac{a+b+\left|a-b\right|}{2}\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy\right\} \\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{-\epsilon}^{\epsilon}f\left(\frac{\max\left(a,b\right)+\min\left(a,b\right)-\left(\max\left(a,b\right)-\min\left(a,b\right)\right)}{2}\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy+\int_{-\epsilon}^{\epsilon}f\left(\frac{\max\left(a,b\right)+\min\left(a,b\right)+\left(\max\left(a,b\right)-\min\left(a,b\right)\right)}{2}\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy\right\} \\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{-\epsilon}^{\epsilon}f\left(\min\left(a,b\right)\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy+\int_{-\epsilon}^{\epsilon}f\left(\max\left(a,b\right)\right)\delta\left(y\right)\frac{1}{\left|a-b\right|}dy\right\} \\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{\min\left(a,b\right)-\epsilon}^{\min\left(a,b\right)+\epsilon}f\left(\min\left(a,b\right)\right)\delta\left(y-\min\left(a,b\right)\right)\frac{1}{\left|a-b\right|}dy+\int_{\max\left(a,b\right)-\epsilon}^{\max\left(a,b\right)+\epsilon}f\left(\max\left(a,b\right)\right)\delta\left(y-\max\left(a,b\right)\right)\frac{1}{\left|a-b\right|}dy\right\} \cmt{y\rightarrow y-\min\left(a,b\right),y\rightarrow y-\max\left(a,b\right)}\\ & =\lim_{\epsilon\rightarrow+0}\left\{ \int_{\min\left(a,b\right)-\epsilon}^{\min\left(a,b\right)+\epsilon}f\left(y\right)\delta\left(y-\min\left(a,b\right)\right)\frac{1}{\left|a-b\right|}dy+\int_{\max\left(a,b\right)-\epsilon}^{\max\left(a,b\right)+\epsilon}f\left(y\right)\delta\left(y-\max\left(a,b\right)\right)\frac{1}{\left|a-b\right|}dy\right\} \\ & =\int_{-\infty}^{\infty}f\left(y\right)\delta\left(y-\min\left\{ a,b\right\} \right)\frac{1}{\left|a-b\right|}dy+\int_{-\infty}^{\infty}f\left(y\right)\delta\left(y-\max\left\{ a,b\right\} \right)\frac{1}{\left|a-b\right|}dy\\ & =\int_{-\infty}^{\infty}f\left(y\right)\frac{\delta\left(y-a\right)+\delta\left(y-b\right)}{\left|a-b\right|}dy \end{align*} 途中の積分範囲で\(\epsilon\)の2次以降は無視しています。
これより、
\[ \delta\left(\left(x-a\right)\left(x-b\right)\right)=\frac{1}{\left|a-b\right|}\left(\delta\left(x-a\right)+\delta\left(x-b\right)\right) \] となるので与式は成り立つ。

(10)

\begin{align*} \int_{-\infty}^{\infty}f\left(x\right)\delta^{\left(n\right)}\left(x\right)dx & =\left[f\left(x\right)\delta^{\left(n-1\right)}\left(x\right)\right]_{-\infty}^{\infty}-\int_{-\infty}^{\infty}f^{\left(1\right)}\left(x\right)\delta^{\left(n-1\right)}\left(x\right)dx\\ & =-\int_{-\infty}^{\infty}f^{\left(1\right)}\left(x\right)\delta^{\left(n-1\right)}\left(x\right)dx\\ & =\left(-1\right)^{n}\int_{-\infty}^{\infty}f^{\left(n\right)}\left(x\right)\delta\left(x\right)dx+\sum_{k=0}^{n-1}\left\{ \int_{-\infty}^{\infty}\left(-1\right)^{k}f^{\left(k\right)}\left(x\right)\delta^{\left(n-k\right)}\left(x\right)dx-\left(-1\right)^{k+1}\int_{-\infty}^{\infty}f^{\left(k+1\right)}\left(x\right)\delta^{\left(n-k-1\right)}\left(x\right)dx\right\} \\ & =\left(-1\right)^{n}\int_{-\infty}^{\infty}f^{\left(n\right)}\left(x\right)\delta\left(x\right)dx \end{align*} これより、
\[ f\left(x\right)\delta^{\left(n\right)}\left(x\right)=\left(-1\right)^{n}f^{\left(n\right)}\left(x\right)\delta\left(x\right) \] となるので与式は成り立つ。

(11)

\begin{align*} x^{m}\delta^{\left(n\right)}\left(x\right) & =\left(-1\right)^{n}\left(x^{m}\right)^{\left(n\right)}\delta\left(x\right)\\ & =\left(-1\right)^{n}P\left(m,n\right)x^{m-n}\delta\left(x\right)\\ & =\left(-1\right)^{n}P\left(m,n\right)\delta_{m,n}\delta\left(x\right)\\ & =\left(-1\right)^{n}n!\delta_{m,n}\delta\left(x\right) \end{align*}

(12)

\begin{align*} \int_{-\infty}^{t}f\left(x\right)\delta\left(x\right)dx & =\begin{cases} f\left(0\right) & 0<t\\ 0 & t<0 \end{cases}\\ & =\begin{cases} f\left(0\right)H_{c}\left(t\right) & 0<t\\ f\left(0\right)H_{c}\left(t\right) & t<0 \end{cases}\\ & =f\left(0\right)H_{c}\left(t\right) \end{align*}