N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3/2 Delta Sequence

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It has been suggested that this page be renamed.
In particular: can't use / in a title for obvs rsns

Suggest rename to 'N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3 Halfs Delta Sequence' --Hbghlyj (talk) 13:59, 10 May 2024 (UTC)suggest rename to an examples/arbitrary example 1 style page like recent examples

Please rename, because I cannot rename page. Thanks --Hbghlyj (talk) 15:36, 12 May 2024 (UTC)
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Theorem

The graph of the $\ds \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2}}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.

Let $\sequence {\map {\delta_n} x}$ be a sequence such that:

$\ds \map {\delta_n} x := \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} }$

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

\(\ds \int_0^\infty \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\) \(=\) \(\ds \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd \paren {n x}\)
\(\ds \) \(=\) \(\ds \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) $n x = y$, Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 2 \intlimits {\frac y {\paren{1 + y^2}^{1 / 2} } } 0 \infty\) by $\ds \frac 1 {\paren{1 + y^2}^{3 / 2} } = \frac \rd {\rd y} \frac y {\paren{1 + y^2}^{1 / 2} }$
\(\ds \) \(=\) \(\ds \frac 1 2\) by $\ds \lim_{y \to \infty} \frac y {\paren{1 + y^2}^{1 / 2} } = 1$

By Definite Integral of Even Function:

\(\ds \int_{-\infty}^\infty \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\) \(=\) \(\ds 1\)

Let $a,b \in \R$. Then:

\(\ds \int_a^b \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\) \(=\) \(\ds \int_a^b \frac 1 2 \frac 1 {\paren{1 + \paren {n x}^2}^{3 / 2} } \rd \paren {n x}\)
\(\ds \) \(=\) \(\ds \int_{n a}^{n b} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) $n x = y$, Integration by Substitution

Suppose $0 < a < b$. Then:

\(\ds \lim_{n \mathop \to \infty} \int_{a n}^{b n} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_0^{b n} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y - \lim_{n \mathop \to \infty} \int_0^{a n} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) Sum of Integrals on Adjacent Intervals for Integrable Functions
\(\ds \) \(=\) \(\ds \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y - \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\)
\(\ds \) \(=\) \(\ds \frac 1 2 - \frac 1 2\)
\(\ds \) \(=\) \(\ds 0\)

Analogously, suppose $a < b < 0$. Then:

\(\ds \lim_{n \mathop \to \infty} \int_{a n}^{b n} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_{a n}^0 \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y - \lim_{n \mathop \to \infty} \int_{b n}^0 \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\) Sum of Integrals on Adjacent Intervals for Integrable Functions
\(\ds \) \(=\) \(\ds \int_{-\infty}^0 \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y - \int_{-\infty}^0 \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y\)
\(\ds \) \(=\) \(\ds \frac 1 2 - \frac 1 2\)
\(\ds \) \(=\) \(\ds 0\)

Let $\epsilon \in \R_{> 0}$. Then:

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map \phi x \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^{-\epsilon} \map \phi x \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x + \lim_{n \mathop \to \infty} \int_{-\epsilon}^\epsilon \map \phi x \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x + \lim_{n \mathop \to \infty} \int_\epsilon^\infty \map \phi x \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\)
\(\ds \) \(=\) \(\ds \map \phi {\xi_-} \lim_{n \mathop \to \infty} \int_{-\infty}^{-\epsilon} \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x + \map \phi {\xi_\epsilon} \lim_{n \mathop \to \infty} \int_{-\epsilon}^\epsilon \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x + \map \phi {\xi_+} \lim_{n \mathop \to \infty} \int_{\epsilon}^\infty \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x\) Mean value theorem for integrals, $\xi_\epsilon \in \closedint {-\epsilon} \epsilon$, $\xi_- \in \hointl {-\infty} {-\epsilon}$, $\xi_+ \in \hointr \epsilon \infty$
\(\ds \) \(=\) \(\ds 0 + \map \phi {\xi_\epsilon} \lim_{n \mathop \to \infty} \int_{-n \epsilon}^{n \epsilon} \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y + 0\)
\(\ds \) \(=\) \(\ds \map \phi {\xi_\epsilon}\)

$\epsilon$ is an arbitrary positive real number.

Hence, for every $\epsilon \in \R_{> 0}$ contributions from expressions with $\map \phi {\xi_+}$ and $\map \phi {\xi_-}$ vanish.

Suppose $\xi_\epsilon \ne 0$.

By Real Numbers are Densely Ordered:

$\forall \epsilon \in \R_{> 0} : \exists \epsilon' \in \R_{> 0} : 0 < \epsilon' < \epsilon$

Then with respect to $\epsilon'$ we have that $\xi_\epsilon = \xi_{+'}$ or $\xi_\epsilon = \xi_{-'}$, where $\xi_{+'} \in \hointr {\epsilon'} \infty$ and $\xi_{-'} \in \hointl {-\infty} {-\epsilon'}$.

But from the result above, for every $\epsilon' \in \R_{> 0}$ contributions from expressions with $\map \phi {\xi_{+'}}$ and $\map \phi {\xi_{-'}}$ vanish.

Therefore, the only nonvanishing contribution can come from $\xi_\epsilon = 0$.

$\blacksquare$

Also see

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