I am trying to find the all the following Fourier Transforms on a finite time interval $[t_0,\ t_f]$ with $t_0<t_f$:
- $$A(iw) = \int\limits_{t_0}^{t_f} f(t)\ \delta(t-t_0)\ e^{-iwt} dt$$
- $$B(iw) = \int\limits_{t_0}^{t_f} f(t)\ \delta(t-t_0)\ e^{-iwt} dt$$
where $f(t)$ is a well behaved function which is not zero necessarily at $\{t_0;\ t_f\}$, and $\delta(t)$ is the Dirac's delta function. Please consider that $\theta(t)$ is Heaviside step function. The idea is express the functions as results dependent on other functions and in terms of $\int\limits_{t_0}^{t_f} f(t)\ e^{-iwt} dt$
I tried to solve it by myself but I think I am doing something wrong, since I find different values of $\theta(0)$ if I try to solve:
- $$X(iw) = \int\limits_{t_0}^{t_f} f(t) \left(\theta(t-t_0)-\theta(t-t_f)\right) e^{-iwt} dt$$
- $$Y(iw) = \int\limits_{t_0}^{t_f} f(t) \theta(t-t_0)\theta(t_f-t) e^{-iwt} dt$$
when conceptually they should be the same. I don't know if I am messing up with Cauchy's principal values, with issues when $0<t_0<t_f$ compared $t_0<0<t_f$, or if something with distribution theory which I don't know. Neither I am sure if $\theta(0)$ could be treated as a constant so it could go inside/outside the Fourier Transforms without making problems.
But otherwise, I do was able to fulfill Parseval's theorem on many examples when using:$$\int\limits_{t_0}^{t_f} f'(t) \left(\theta(t-t_0)-\theta(t-t_f)\right) e^{-iwt} dt:= iw \int\limits_{t_0}^{t_f} f(t)\ e^{-iwt} dt + f(t_f)e^{-iwt_f}-f(t_0)e^{-iwt_0}$$which I found using my flawed formulas, but if I made the direct replacement $f'(t)\to f(t)$ and $f(t) \to F(t)=\int_{t_0}^t f(u)\ du$ it stops working, and I think it is related.
I hope you could help me to find the formulas for 1 and 2 (if you could go 1-4 even better): If they change if I consider instead the interval $(t_0,\ t_f)$ please show them also.
I hope you could explain if I have something conceptually wrong about the things I mentioned too.