I don't like doing this, for some reason, but it seems I gotta get all the help I have available to get those courses out of my back.
I 'll be asking some S&S related questions here, have a look at them when you get the time:
Q1: Prove that δ'(-t)=-δ'(t)
As hinted, I attempted this by using \(\int \phi (t) f_1(t) dt =\int \phi (t) f_2(t) dt \Leftrightarrow f_1(t)=f_2(t)\)
So,
\(\int_{-\infty}^{\infty} \phi(t) (- \delta ' (t)) dt=\int_{-\infty}^{\infty} \phi '(t) \delta (t) dt=\phi'(0)\)
and
\(\int_{-\infty}^{\infty} \phi(t) \delta ' (-t) dt=-\int_{\infty}^{-\infty} \phi(-\tau) \delta ' (\tau) d\tau\)
but I can't see the last step, to get to the same result. Any hints?
I 'll be asking some S&S related questions here, have a look at them when you get the time:
Q1: Prove that δ'(-t)=-δ'(t)
As hinted, I attempted this by using \(\int \phi (t) f_1(t) dt =\int \phi (t) f_2(t) dt \Leftrightarrow f_1(t)=f_2(t)\)
So,
\(\int_{-\infty}^{\infty} \phi(t) (- \delta ' (t)) dt=\int_{-\infty}^{\infty} \phi '(t) \delta (t) dt=\phi'(0)\)
and
\(\int_{-\infty}^{\infty} \phi(t) \delta ' (-t) dt=-\int_{\infty}^{-\infty} \phi(-\tau) \delta ' (\tau) d\tau\)
but I can't see the last step, to get to the same result. Any hints?