Editing Dirac delta function (section)

===Composition with a function===
More generally, the delta distribution may be [[distribution (mathematics)#Composition with a smooth function|composed]] with a smooth function {{math|''g''(''x'')}} in such a way that the familiar change of variables formula holds (where <math>u=g(x)</math>), that

<math display="block">\int_{\R} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left|g'(x)\right| dx = \int_{g(\R)} \delta(u)\,f(u)\,du</math>

provided that {{mvar|g}} is a [[continuously differentiable]] function with {{math|''g&prime;''}} nowhere zero.{{sfn|Gelfand|Shilov|1966–1968|loc=Vol. 1, §II.2.5}} That is, there is a unique way to assign meaning to the distribution <math>\delta\circ g</math> so that this identity holds for all compactly supported test functions {{mvar|f}}.  Therefore, the domain must be broken up to exclude the {{math|1=''g&prime;'' = 0}} point.  This distribution satisfies {{math|1=''δ''(''g''(''x'')) = 0}} if {{mvar|g}} is nowhere zero, and otherwise if {{mvar|g}} has a real [[root of a function|root]] at {{math|''x''<sub>0</sub>}}, then

<math display="block">\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.</math>

It is natural therefore to {{em|define}} the composition {{math|''δ''(''g''(''x''))}} for continuously differentiable functions {{mvar|g}} by

<math display="block">\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}</math>

where the sum extends over all roots of {{mvar|''g''(''x'')}}, which are assumed to be [[simple root|simple]].  Thus, for example

<math display="block">\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|} \Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].</math>

In the integral form, the generalized scaling property may be written as

<math display="block"> \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}. </math>