狄拉克δ函数

定义

$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$

$\int_{-\infty}^\infty \delta(x) \, dx = 1.$

$\delta(x-c) = \begin{cases} +\infty, & x = c \\ 0, & x \ne c \end{cases}$
$\int_a^b\delta(x-c)dx=1,\quad a

性质

缩放和对称

$\int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}$

$\delta(\alpha x) = \frac{\delta(x)}{|\alpha|}.$

$\delta(-x) = \delta(x)$

代数性质

δ与x分布积等于零：

$x\delta(x) = 0.$

$f(x) = g(x) +c \delta(x)$

c。为常数[8]

平移

$\int_{-\infty}^\infty f(t) \delta(t-T)\,dt = f(T).$

 $(f(t) * \delta(t-T))\,$ $\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau) \delta(t-T-\tau) \, d\tau$ $= \int\limits_{-\infty}^\infty f(\tau) \delta(\tau-(t-T)) \, d\tau$ （运用$\delta(-x)=\delta(x)$） $= f(t-T).\,$

$\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \, dx = \delta(\xi-\eta).$

其他性质

• $\delta^\prime(-x)=-\delta^\prime(x)$
• $\delta(x^2-a^2)=\frac{1}{2|a|}[\delta(x+a)+\delta(x-a)]$
• $\int_a^b f(x)\delta(x-c)dx=f(c),\quad a
• $\int_a^b f(x)\frac{d^n}{dx^n}\delta(x-c)dx=(-1)^n\left[\frac{d^n}{dx^n}f(x)\right]_{x=c},\quad a

傅立叶变换

δ函数是一个缓增广义函数。因此，它的傅里叶变换是良好定义的。正式地，

$\hat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi}\delta(x)\,dx = 1.$

δ函数的表达式

$\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), \,$

$\lim_{\varepsilon\to 0^+} \int_{-\infty}^{\infty}\eta_\varepsilon(x)f(x) \, dx = f(0) \$

• $\delta(x)=\lim_{\alpha\to 0^+}\frac{1}{\pi}\frac{\alpha}{\alpha^2+x^2}$
• $\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{ikx}dk$
• $\delta(x)=\lim_{k\to\infty}\frac{1}{\pi}\frac{\sin{kx}}{x}$

注释

1. ^ Dirac 1958，§15 The δ function, p. 58
2. ^ Gel'fand & Shilov 1968，Volume I, §§1.1, 1.3
3. ^ Schwartz 1950，第3页
4. ^ Arfken & Weber 2000，第84页
5. ^ Bracewell 1986，Chapter 5