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Dirac delta and exponential

  1. Aug 4, 2006 #1
    Let be the exponential:

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

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

    [tex] \delta (x) \sim \frac{sin(nx)}{\pi x} [/tex] then differentiating..

    [tex] \delta ' (x) \sim \frac{ncos(nx)}{\pi x}- \frac{\delta (x)}{\pi x} [/tex]

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

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

    thanks....:rolleyes: :grumpy: :tongue2:
  2. jcsd
  3. Aug 10, 2006 #2
    You can say:

    [tex]\frac{\sin(nx)}{\pi x} \rightarrow \delta (x) [/tex]

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

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

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