N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence/Proof 2
Theorem
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
- $\ds \map {\delta_n} x := \frac n \pi \frac 1 {1 + n^2 x^2}$
Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence.
That is, in the distributional sense it holds that:
- $\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$
or
- $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x = \map \delta \phi$
where $\phi \in \map \DD \R$ is a test function, $\delta$ is the Dirac delta distribution, and $\map \delta x$ is the abuse of notation, usually interpreted as an infinitely thin and tall spike with its area equal to $1$.
Proof
Let $\map g x = \map \phi x - \map \phi 0$.
Then:
- $\ds \int_{- \infty}^\infty \map \phi x \map {\delta_n} x \rd x = \map \phi 0 + \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x$
Let $A \in \R_{> 0}$.
Then:
\(\ds \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x\) | \(=\) | \(\ds \int_{- \infty}^{- A} \map g x \map {\delta_n} x \rd x + \int_A^\infty \map g x \map {\delta_n} x \rd x + \int_{- A}^A \map g x \map {\delta_n} x \rd x\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
\(\ds \) | \(=:\) | \(\ds I_1 + I_2 + I_3\) |
Let:
- $\ds \max_{x \mathop \in \closedint {-A} A} \size {\map g x} := \map M A$
Then:
\(\ds I_3\) | \(\le\) | \(\ds \int_{-A}^A \size {\map g x} \map {\delta_n} x \rd x\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \map M A \int_{-A}^A \frac n {\pi \paren {1 + n^2 x^2} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map M A \frac 2 \pi \map \arctan {n A}\) | Primitive of Reciprocal of x squared plus a squared | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map M A\) | $\ds \forall x \in \R_{\ge 0} : 0 \le \map \arctan x \le \frac \pi 2$ |
We have that $\map g 0 = 0$.
By definition, $\phi$ is a smooth real function on $\R$.
By Differentiable Function is Continuous, $\map g x$ is continuous at $x = 0$.
Furthermore:
\(\ds \lim_{A \mathop \to 0} \map M A\) | \(=\) | \(\ds \lim_{A \mathop \to 0} \max_{x \mathop \in \closedint {-A} A} \size {\map g x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size {\map g 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
By definition of the limit of a real function:
- $\forall \epsilon' \in \R_{>0} : \exists \delta \in \R_{>0}: \forall A \in \R_{> 0}: 0 < A < \delta \implies \map M A < \epsilon'$
Let $\ds \epsilon' = \frac \epsilon 2$.
It follows that:
- $\ds \forall \epsilon \in \R_{> 0} : \exists A \in \R_{> 0} : I_3 \le \map M A < \frac \epsilon 2$
Suppose $A$ is such that the above inequality holds.
By definition, $\map \phi x$ is bounded.
Then:
\(\ds \size {\map g x}\) | \(=\) | \(\ds \size {\map \phi x - \map \phi 0}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \size {\map \phi x} + \size {\map \phi 0}\) | Triangle Inequality for Real Numbers |
It follows that:
- $\exists b \in \R_{> 0} : \forall x \in \R : \size {\map g x} < b$
Then:
\(\ds \size {I_1 + I_2}\) | \(=\) | \(\ds \size { \int_{- \infty}^{-A} \map g x \map {\delta_n} x \rd x + \int_A^\infty \map g x \map {\delta_n} x \rd x}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {\int_{- \infty}^{-A} \size {\map g x} \map {\delta_n} x \rd x + \int_A^\infty \size {\map g x} \map {\delta_n} x \rd x}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size { b \int_{- \infty}^{-A} \map {\delta_n} x \rd x + b \int_A^\infty \map {\delta_n} x \rd x}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds b \size {\int_{- \infty}^{-A} \map {\delta_n} x \rd x + \int_A^\infty \map {\delta_n} x \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac b \pi \size {\bigintlimits {\map \arctan {n x} } {- \infty} {-A} + \bigintlimits {\map \arctan {n x} } A \infty }\) | Primitive of Reciprocal of x squared plus a squared | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac b \pi \size {- \map \arctan {n A} + \frac \pi 2 + \frac \pi 2 - \map \arctan {n A} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b \size {1 - \frac 2 \pi \map \arctan {n A} }\) |
With the number $A$ fixed:
- $\ds \lim_{n \mathop \to \infty} \frac 2 \pi \map \arctan {n A} = 1$.
By Squeeze Theorem and for a fixed $A$ we have:
- $\ds \lim_{n \mathop \to \infty} \size {I_1 + I_2} = 0$
By definition of the limit of a real sequence:
- $\ds \forall \overline \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \size {I_1 + I_2} \le b \size {1 - \frac 2 \pi \map \arctan {n A} } < \overline \epsilon$.
Let $\ds \overline \epsilon = \frac \epsilon 2$.
Then:
- $\ds \forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \size {I_1 + I_2} < \frac \epsilon 2$.
Let $A$ and $N$ be such that the above inequalities for $I_3$ and $I_1 + I_2$ hold.
Then:
\(\ds \size {\int_{-\infty}^\infty \map g x \map {\delta_n} x \rd x}\) | \(\le\) | \(\ds \size {I_1 + I_2 + I_3}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {I_1 + I_2} + \size {I_3}\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
To sum up:
- $\ds \forall \epsilon \in \R_{>0} : \exists N \in \R_{>0} : \forall n \in \N_{>0} : \forall n > N \implies \size {\int_{-\infty}^\infty \map g x \map {\delta_n} x \rd x} < \epsilon$
By definition of the limit of a real sequence:
- $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map g x \map {\delta_n} x \rd x = 0$
However:
\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map g x \map {\delta_n} x \rd x\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map \phi x \map {\delta_n} x \rd x - \map \phi 0 \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \rd x\) | Sum Rule for Limits of Real Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map \phi x \map {\delta_n} x \rd x - \map \phi 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map \phi x \map {\delta_n} x \rd x\) | \(=\) | \(\ds \map \phi 0\) |
$\blacksquare$
Sources
2013: George Arfken, Hans J. Weber and Frank E. Harris: Mathematical Methods for Physicists (7th ed.): Chapter $1$ Mathematical Preliminaries $1.11$ Dirac Delta Function