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Tbonewillsone
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Homework Statement
Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I understand this might not be possible, and if so I'd like to know why!
Homework Equations
If we have
\begin{equation}
I= \int^{\infty}_{-\infty} \lim_{n \rightarrow \infty}\left( \frac{\sin (n x)}{x} \right) f(x) \mathrm dx=\lim_{n\rightarrow \infty}I_{n},
\end{equation}
then through a change of variables, nx=y
\begin{equation}
I_{n} = \int^{\infty}_{-\infty} \left( \frac{\sin (n x)}{x} \right) f(x) \mathrm dx = \int^{\infty}_{-\infty} \left( \frac{\sin (y)}{y} \right) f \left(\frac{y}{n} \right) \mathrm dy,
\end{equation}
and so
\begin{equation}
I = \int^{\infty}_{-\infty}\left( \frac{\sin (y)}{y} \right) f \left( 0 \right) \mathrm dy = \pi f \left( 0 \right) .
\end{equation}
This replicates the Dirac delta function, meaning that at this limit we can say
\begin{equation}
\lim_{n \rightarrow \infty} \left( \frac{\sin (n x)}{x} \right) \rightarrow \pi \delta(x).
\end{equation}
The Attempt at a Solution
I would like to expand the test function, and then, through taking the limit, recover the delta function property of our function.
\begin{equation}
I_{n} = \int^{\infty}_{-\infty} \left( \frac{\sin (n x)}{x} \right) f(x) \mathrm dx = \int^{\infty}_{-\infty} \left( \frac{\sin (y)}{y} \right) f \left(\frac{y}{n} \right) \mathrm dy,
\end{equation}
Putting the function through a Taylor expansion,
\begin{equation}
\int^{\infty}_{-\infty} \left( \frac{\sin (y)}{y} \right) f \left(\frac{y}{n} \right) \mathrm dy= \int^{\infty}_{-\infty} \left( \frac{\sin (y)}{y} \right) \left( f \left(0\right) + \frac{y}{n}f^{\prime} \left(0\right) + \frac{y^{2}}{n^{2}}f^{\prime \prime} \left( 0 \right) + \mathcal O\left( \frac{y^{3}}{n^{3}} \right) \right) \mathrm dy,
\end{equation}
This integral clearly diverges, I could take the limit at this stage, but I don't understand why this expansion would not work, it must be somehow linked to how "integrateable" the sin(nx)/x is. Is there any way I can manipulate the taylor series to get some converging terms together?
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