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.