Definition:Dirac Delta Function/Also defined as

Dirac Delta Function: Also defined as

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The Dirac delta function is also defined by the following limits:

\(\text {(1)}: \quad\)

\(\ds \map \delta x\)

\(=\)

\(\ds \dfrac 1 \pi \lim_{\epsilon \mathop \to 0} \dfrac \epsilon {x^2 + \epsilon^2}\)

\(\text {(2)}: \quad\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \lim_{\epsilon \mathop \to 0} \epsilon \size x^{\epsilon - 1}\)

\(\text {(3)}: \quad\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt \pi} \lim_{\epsilon \mathop \to 0} \dfrac 1 {\sqrt {4 \epsilon} } e^{-x^2 / {4 \epsilon} }\)

\(\text {(4)}: \quad\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\pi x} \lim_{\epsilon \mathop \to 0} \map \sin {\dfrac x \epsilon}\)

Sources

• Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeltaFunction.html