Dirac delta

The Dirac delta function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x) } satisfies the following property:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^\infty f(x)\delta(x-y)dx = f(y)}

for all functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} . This can be seen as the continuous analogue of Kronecker Delta. Taking the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=1} for all $x$, using the above property and letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^\infty \delta(x)dx = 1}

Therefore, the Dirac delta function is normalized.

Technically speaking, no function satisfies the first property above. However, the notion of Dirac delta can be made mathematically rigorous. In practice, it can be seen as the limit of a function which becomes extremely concentrated at a single point. Thus it is often said that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x) = \begin{cases} \infty & \mbox{if } x = 0 \\ 0 & \mbox{if } x \ne 0 \end{cases}}

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} the Dirac delta function can be generalized to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbb{R}^n}f(\mathbf{x})\delta(\mathbf{x}-\mathbf{y})d^n\mathbf{x} = f(\mathbf{y})}

References

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