I am having some trouble trying to evaluate the integral of this F.T.

We are to take the definition for a continuous F.T:

$$ \mathcal{F} [g(t)] = G(f) = \int_{-\infty}^{\infty} (e^{-i2\pi ft})g(t) dt $$

Where $g(t)$ is the an exponential function: $g(t) = e^{-t/\tau} H(t)$

Where H(t) is the Heaviside step function and is defined as:

$$ H(t) = \left\{\begin{array}{ll} 1, & t \gt 0 \\ 0, & t \le 0 \end{array}\right. $$

I have an initial attempt of the integral worked out but upon evaluation I get an answer that is undefined. I appreciate the help!


For $\tau>0$, we have

$$\begin{align} \mathscr{F}\{g\}(f)&=\int_0^\infty e^{-t/\tau}e^{-i2\pi ft }\,dt\\\\ &=\lim_{L\to \infty}\left.\left(\frac{e^{-(\frac1\tau+i2\pi f)t}}{-(\frac1\tau+i2\pi f)}\right)\right|_{t=0}^{t=L}\\\\ &=\frac{1}{\frac1\tau+i2\pi f} \end{align}$$


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