# Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 1

## Theorem

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a delta sequence.

Let $f : \R \to \R$ be a real function with a simple zero at $x_0$.

Let $f$ be strictly monotone.

Let $\phi \in \map \DD \R$ be a test function.

Then in the distributional sense it holds that:

$\ds \map \delta {\map f x} = \frac {\map \delta {x - x_0}}{\size {\map {f'} {x_0}} }$

which can be interpreted as:

$\ds \int_{-\infty}^\infty \map \delta {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$

which more strictly means that:

$\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$

## Proof

Suppose $\map {f'} {x_0} > 0$.

Then:

 $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} {\map f x} \map \phi x \rd x$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} y \frac {\map \phi {\map x y} }{\map {f'} {\map x y} } \rd y$ Derivative of Inverse Function, Integration by Substitution, $y = \map f x$ $\ds$ $=$ $\ds \frac {\map \phi {x_0} }{\map {f'} {x_0} }$ Definition of Delta Sequence, $\map x 0 = x_0$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{\map {f'} {x_0} } \rd x$

Suppose $\map {f'} {x_0} < 0$.

Then:

 $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} {\map f x} \map \phi x \rd x$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{\infty}^{-\infty} \map {\delta_n} y \frac {\map \phi {\map x y} }{\map {f'} {\map x y} } \rd y$ Derivative of Inverse Function, Integration by Substitution, $y = \map f x$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} y \frac {\map \phi {\map x y} }{- \map {f'} {\map x y} } \rd y$ $\ds$ $=$ $\ds \frac {\map \phi {x_0} }{- \map {f'} {x_0} }$ Definition of Delta Sequence, $\map x 0 = x_0$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{- \map {f'} {x_0} } \rd x$

Altogether:

 $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} {\map f x} \map \phi x \rd x$ $=$ $\ds \frac {\map \phi {x_0} }{ \size { \map {f'} {x_0} } }$ Definition of Delta Sequence, $\map x 0 = x_0$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{ \size {\map {f'} {x_0 } } } \rd x$

$\blacksquare$

## Sources

1998: Ram Prakash Kanwal: Generalized Functions: Theory and Technique (2nd ed.): Chapter $3$. Additional Properties of Distributions $3.1$ Transformation Properties of the Delta Distribution