linguistic:translation
Composition with a function

More generally, the delta distribution may be composed with a smooth function g (x) in such a way that the familiar change of variables formula holds, that

∫ R δ (g (x)) f (g (x)) | g ′ (x) | d x = ∫ g (R) δ (u) f (u) d u {\displaystyle \int _{\mathbb {R} }\delta {\bigl (}g(x){\bigr)}f{\bigl (}g(x){\bigr)}\left|g'(x)\right|dx=\int _{g(\mathbb {R})}\delta (u)\,f(u)\,du}

provided that g is a continuously differentiable function with g ′ nowhere zero. That is, there is a unique way to assign meaning to the distribution δ ∘ g {\displaystyle \delta \circ g} so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g ′ = 0 point. This distribution satisfies δ (g (x)) = 0 if g is nowhere zero, and otherwise if g has a real root at x, then

δ (g (x)) = δ (x − x 0) | g ′ (x 0) |. {\displaystyle \delta (g(x))={\frac {\delta (x-x_{0})}{|g'(x_{0})|}}.}

It is natural therefore to define the composition δ (g (x)) for continuously differentiable functions g by

δ (g (x)) = ∑ i δ (x − x i) | g ′ (x i) | {\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}

where the sum extends over all roots of g (x), which are assumed to be simple. Thus, for example

δ (x 2 − α 2) = 1 2 | α |. {\displaystyle \delta \left(x^{2}-\alpha ^{2}\right)={\frac {1}{2|\alpha |}}{\Big }.}

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

∫ − ∞ ∞ f (x) δ (g (x)) d x = ∑ i f (x i) | g ′ (x i) |. {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}.}