- #1
kq6up
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Homework Statement
Delta functions said to live under the integral signs, and two expressions (##D_1(x)## and ##D_2(x)##) involving delta functions are said to be equal if:
##\int _{ -\infty }^{ \infty }{ f(x)D_{ 1 }(x)dx } =\int _{ -\infty }^{ \infty }{ f(x)D_{ 2 }(x)dx }##
(a) Show that:
##\delta (cx)=\frac{ 1 }{ |c| } \delta (x)##
Where ##c## is a real constant. (Be sure to check the case where ##c## is negative.)
Homework Equations
Posted above.
The Attempt at a Solution
Let ##u=cx## ##\therefore## ##\frac{1}{c}du=dx##
This yields:
##\frac{1}{c}\int_{-\infty}^{\infty}{f(u/c)\delta (u)du}=\frac{1}{|c|}\int_{-\infty}^{\infty}{f(x)\delta (x)dx}##. This works for a test case where ##c > 0##, but obviously fails when ## c < 0##. I am not sure where this absolute value came from.
Hints please.
Thanks,
Chris