디락 델타 함수 (Dirac delta function)

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References and Related Articles

* Ref. [01] Book - Classical Electro-Dynamics, 3rd Edition, Page 26, Wiley Inc. by John David Jackson.

References and Related Articles

* Ref. [02] 전파거북이's blog - 디랙 델타 함수 (Dirac delta function)

References and Related Articles

* Ref. [03] Wiki - Dirac delta function

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Dirac Delta Function

(x-1)
\[ \delta (x - a) = \begin{cases} \infty &\text{when } x = a \\ 0 &\text{when } x \neq a . \end{cases} \]
(x-2)
\[ \int_{R} d x ~ \delta (x - a) = \begin{cases} 1 &\text{when } a \in R \\ 0 &\text{when } a \notin R . \end{cases} \]
In other words,
(x-3)
\[ \begin{split} \int_{b}^{c} d x ~ \delta (x - a) = 1 \quad\text{and}\quad \int_{c}^{b} d x ~ \delta (x - a) = -1 \end{split} \]
when \(b < a < c\).
(x-4)
\[ \int_{R} d x ~ f(x) \delta (x - a) = \begin{cases} f(a) &\text{when } a \in R \\ 0 &\text{when } a \notin R . \end{cases} \]
(x-5)
\[ \begin{split} &\int_{b}^{c} d x ~ f(x) \frac{d}{d x} \big[ \delta (x - a) \big] \equiv \int_{b}^{c} d x ~ f(x) \delta' (x - a) \\ &= f(x) \delta (x-a) \bigg|_{b}^{c} - \int_{b}^{c} d x ~ \frac{d}{d x} \big[ f(x) \big] \delta (x - a) . \end{split} \]
When \( b<a<c \), therefore
(x-6)
\[ \int_{b}^{c} d x ~ f(x) \delta' (x - a) = - f' (a) . \]
(x-7)
\[ \delta \big( f (x) \big) = \sum_i \frac{1}{ \Big| \frac{d f}{d x} (x_i) \Big|} \delta (x - x_i) \]
where \(x_i\)'s are simple zeros of \(f(x)\).
cf.) \( \delta(-x) = \delta(x) \) and \(\delta(a-x) = \delta(x-a)\).
(x-8)
\[ \delta^3 (\vec{x} - \vec{y}) = \delta (x_1 - y_1) ~ \delta (x_2 - y_2) ~ \delta (x_3 - y_3) \]
in Cartesian coordinate representation.
(x-9)
\[ \int_{R} d \vec{x} ~ \delta^3 (\vec{x} - \vec{y}) = \begin{cases} 1 &\text{when } \vec{y} \in R \\ 0 &\text{when } \vec{y} \notin R . \end{cases} \]
Since
(x-10)
\[ \int_{R} d x ~ d y ~ d z ~ \delta^3 (\vec{x} - \vec{x}') \quad \rightarrow \quad \int_{R} d u ~ d v ~ d w ~ \sqrt{g} ~ \delta^3 (\vec{x} - \vec{x}') , \]
(x-11)
\[ \delta^3 (\vec{x} - \vec{x}') = \frac{1}{\sqrt{g}} \delta (u - u') ~ \delta (v - v') ~ \delta (w - w') . \]

TRefs.References and Related Articles

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1. Ref. [01] Book - Classical Electro-Dynamics, 3rd Edition, Page 26, Wiley Inc. by John David Jackson.
2. Ref. [02] 전파거북이's blog - 디랙 델타 함수 (Dirac delta function)
3. Ref. [03] Wiki - Dirac delta function

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