I was wondering what happens when calculating the fourier transform of a fonction, the limit does not exist.
Let me explain this with an example :
I have the function difined by :
f(t) = e^t when -infinity < t < 0
We then calcul the fourier transform :
F(jw) = integral (-infinity -> 0) f(t) e^-jwt dt
F(jw) = integral (-infinity -> 0) e^(1-jw)t dt
F(jw) = [e^(1-jw)t / (1-jw)] between 0 and -infinity.
Here we can't calcul the limit in -infinity. Can we say that :
F(jw) = 1 / (1-jw) ?
If it is wrong, why ? And what can we do ?
(If it can help, the original question is to calcul the fourier transform of the function f(t) defined by e^t between -infinity and 0 AND e^-t between 0 and infinity).
Thank you guys,
PS : I don't know if there is a proper formating of mathematic formulas, sorry about that if it does :(.
The integral $$ \int_{-\infty}^{0}e^{(1-j\omega) t}dt $$ is defined as the limit of the integral $$ \int_{b}^{0}e^{(1-j\omega) t}dt $$ as $b\rightarrow-\infty$. As you said yourself, these can be calculated by a primitive function:
$$ \int_{b}^{0}e^{(1-j\omega) t}dt=\Bigg[\frac{e^{(1-j\omega)t}}{1-j\omega} \Bigg]_{t=b}^{0}= \frac{1-e^{(1-j\omega)b}}{1-j\omega} $$
As $b\rightarrow-\infty$, the term $e^{(1-j\omega)b}$ tends to $0$, as: $$ |e^{(1-j\omega)b}|=|e^b| |e^{-j\omega t}|=e^b\overset{b\rightarrow-\infty}\longrightarrow0 $$
So we get:
$$ \int_{-\infty}^{0}e^{(1-j\omega) t}dt=\lim_{b\rightarrow-\infty}\int_{b}^{0}e^{(1-j\omega) t}dt=\lim_{b\rightarrow-\infty} \frac{1-e^{(1-j\omega)b}}{1-j\omega}=\frac{1}{1-j\omega} $$