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w**a
发帖数: 1024 | 1 in physics, we know:
the integration of \exp{i*k*x}dx, i=\sqrt{-1}, on the real line from -\infty
to +\infty
is a delta function I=2*\pi*\delta(k).
The usual way to show this identity is to use
Fourier transform of a delta function then study its inverse transform.
Here is my question: integrate the same integrand, but from x=0 to +\infty.
Shall we get one-half of I (above value)? How to show this? I really want to
use distribution technique to show this. Thank you. | s**********n
发帖数: 1485 | 2 So you are asking for a distribution whose Fourier transform is 0 for x<0
and 1 for x>0?
The answer should be (modulo a constant factor)
(i/2)H + (1/2)delta
where H is the Hilbert transform. |
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