Are the Dirac Delta and Exponential Approximations Valid for Large n?

[eljose]

Let be the exponential:

e^{inx}=cos(nx)+isin(nx) n\rightarrow \infty

Using the definition (approximate ) for the delta function when n-->oo

\delta (x) \sim \frac{sin(nx)}{\pi x} then differentiating..

\delta ' (x) \sim \frac{ncos(nx)}{\pi x}- \frac{\delta (x)}{\pi x}

are this approximations true for big "n" ??.. i would like to know this to compute integrals (for big n ) of the form:

\int_{2}^{\infty} dx f(x,n)e^{inx}

thanks...

 

[WigneRacah]

You can say:

\frac{\sin(nx)}{\pi x} \rightarrow \delta (x)

for n \rightarrow \infty in the sense of distributions. But, by derivating, you have:

\frac{n x \cos(n x) - \sin(n x)}{\pi x^2} \rightarrow \delta'(x).

\frac{ \cos (n x)}{x^2} and \frac{1}{x} \delta(x) are not distributions! They are meaningless as distributions.