How do I apply integration by part to an improper integral?
I'm condidering an integral
$$\int^\infty_{-\infty} g(t)f(t)dt$$ so I guess I can do
$$\lim_{b\rightarrow -\infty}\int^0_{b} g(t)f(t)dt + \lim_{a\rightarrow \infty}\int^a_{0} g(t)f(t)dt $$ and then consider these summands one at a time. Such that
$$\lim_{a\rightarrow \infty}\int^a_{0} g(t)f(t)dt = \lim_{a \rightarrow \infty}\left( [g(t)F(t)]_0^a - \int_0^a g'(t)F(t)dt\right),$$
where $f(t) = \frac{d}{dt}F(t)$?
And a follows up question: If I then consider
$$\lim_{a \rightarrow \infty}[g(t)F(t)]_0^a = g(0)F(0) - \lim_{a \rightarrow \infty} f(a)F(a)$$ and $$\lim_{b \rightarrow -\infty}[g(t)F(t)]_b^0 = \lim_{b \rightarrow -\infty} f(b)F(b) - g(0)F(0)$$
and I have the condition that
$$\lim_{b \rightarrow -\infty} f(b)F(b) = \lim_{a \rightarrow \infty} f(a)F(a)$$
does it then follows that
$$\int^\infty_{-\infty} g(t)f(t)dt = - \int_{-\infty}^\infty g'(t) F(t) dt$$ ?
The general formula for proper integrals can be written as
$$ \left. \int_a^b g(t) f(t)\; dt = g(t) F(t)\right|_{t=a}^b - \int_a^b g'(t) F(t)\; dt $$
So if the right side has a limit as $a \to -\infty$ and $b \to +\infty$, then the left side has the same limit.