Integral of Dirac Delta Function by Continuous Function over Reals
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Theorem
Let $\map \delta x$ denote the Dirac delta function.
Let $g$ be a continuous real function.
Then:
- $\ds \int_{- \infty}^{+ \infty} \map \delta x \map g x \rd x = \map g 0$
Proof
We have that:
- $\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
- $\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
\(\ds \int_{- \infty}^{+ \infty} \map {F_\epsilon} x \map g x \rd x\) | \(=\) | \(\ds \int_{-\infty}^{-\epsilon} 0 \times \map g x \rd x + \int_{-\epsilon}^\epsilon \dfrac 1 {2 \epsilon} \map g x \rd x + \int_\epsilon^\infty 0 \times \map g x \rd x\) | Definition 2 of Dirac Delta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to -\infty} \int_L^{-\epsilon} 0 \times \map g x \rd x + \int_{-\epsilon}^\epsilon \dfrac 1 {2 \epsilon} \, \map g x \rd x + \lim_{L \mathop \to \infty} \int_\epsilon^L 0 \times \map g x \rd x\) | Definition of Improper Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to -\infty} \int_L^{-\epsilon} 0 \rd x + \dfrac 1 {2 \epsilon} \int_{-\epsilon}^\epsilon \map g x \rd x + \lim_{L \mathop \to \infty} \int_\epsilon^L 0 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to -\infty} \paren {0 \times \paren {-\epsilon - L} } + \dfrac 1 {2 \epsilon} \int_{-\epsilon}^\epsilon \map g x \rd x + \lim_{L \mathop \to \infty} \paren {0 \times \paren {L - \epsilon} }\) | Definite Integral of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to -\infty} 0 + \dfrac 1 {2 \epsilon} \int_{-\epsilon}^\epsilon \map g x \rd x + \lim_{L \mathop \to \infty} 0\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 \epsilon} \int_{-\epsilon}^\epsilon \map g x \rd x\) |
From Darboux's Theorem:
- $\ds m \paren {\epsilon - \paren {-\epsilon} } \le \int_{-\epsilon}^\epsilon \map g x \rd x \le M \paren {\epsilon - \paren {-\epsilon}}$
where:
on $\closedint {-\epsilon} \epsilon$.
Hence:
- $\ds 2 m \epsilon \le \int_{-\epsilon}^\epsilon \map g x \rd x \le 2 M \epsilon$
and so dividing by $2 \epsilon$:
- $\ds m \le \dfrac 1 {2 \epsilon} \int_{-\epsilon}^\epsilon \map g x \rd x \le M$
Then:
- $\ds \lim_{\epsilon \mathop \to 0} M = m = \map g 0$
and so by the Squeeze Theorem:
- $\ds \lim_{\epsilon \mathop \to 0} \int_{-\infty}^{+\infty} \map {F_\epsilon} x \map g x \rd x = \map g 0$
But by Definition 2 of Dirac Delta Function:
- $\ds \lim_{\epsilon \mathop \to 0} \int_{-\infty}^{+\infty} \map {F_\epsilon} x \map g x \rd x = \ds \int_{-\infty}^{+\infty} \map \delta x \map g x \rd x$
Hence the result.
$\blacksquare$