디락 델타 함수 (Dirac delta function)

Dirac Delta Function

\delta (x - a) = \begin{cases} \infty &\text{when } x = a \\ 0 &\text{when } x \neq a . \end{cases}
\int_{R} d x ~ \delta (x - a) = \begin{cases} 1 &\text{when } a \in R \\ 0 &\text{when } a \notin R . \end{cases} In other words, \begin{split} \int_{b}^{c} d x ~ \delta (x - a) = 1 \quad\text{and}\quad \int_{c}^{b} d x ~ \delta (x - a) = -1 \end{split} when \(b < a < c\).
\int_{R} d x ~ f(x) \delta (x - a) = \begin{cases} f(a) &\text{when } a \in R \\ 0 &\text{when } a \notin R . \end{cases}
\begin{split} &\int_{b}^{c} d x ~ f(x) \frac{d}{d x} \big[ \delta (x - a) \big] \equiv \int_{b}^{c} d x ~ f(x) \delta' (x - a) \\ &= f(x) \delta (x-a) \bigg|_{b}^{c} - \int_{b}^{c} d x ~ \frac{d}{d x} \big[ f(x) \big] \delta (x - a) . \end{split} When \( b<a<c \), therefore \int_{b}^{c} d x ~ f(x) \delta' (x - a) = - f' (a) .
\delta \big( f (x) \big) = \sum_i \frac{1}{ \Big| \frac{d f}{d x} (x_i) \Big|} \delta (x - x_i) where \(x_i\)'s are simple zeros of \(f(x)\).
cf.) \( \delta(-x) = \delta(x) \) and \(\delta(a-x) = \delta(x-a)\).
\delta^3 (\vec{x} - \vec{y}) = \delta (x_1 - y_1) ~ \delta (x_2 - y_2) ~ \delta (x_3 - y_3) in Cartesian coordinate representation.
\int_{R} d \vec{x} ~ \delta^3 (\vec{x} - \vec{y}) = \begin{cases} 1 &\text{when } \vec{y} \in R \\ 0 &\text{when } \vec{y} \notin R . \end{cases} Since \int_{R} d x ~ d y ~ d z ~ \delta^3 (\vec{x} - \vec{x}') \quad \rightarrow \quad \int_{R} d u ~ d v ~ d w ~ \sqrt{g} ~ \delta^3 (\vec{x} - \vec{x}') , \delta^3 (\vec{x} - \vec{x}') = \frac{1}{\sqrt{g}} \delta (u - u') ~ \delta (v - v') ~ \delta (w - w') .

References and Related Articles

1. Book - Classical Electro-Dynamics, 3rd Edition, Page 26, Wiley Inc. by John David Jackson.
2. 전파거북이's blog - 디랙 델타 함수 (Dirac delta function)
3. Wiki - Dirac delta function