MHB Fourier Transform limits problem

[ognik]

Find the Fourier Transform of $ e^{-a|t|}Cosbt $

I'd like to simplify this using $Cosbt = Re\left\{e^{ibt}\right\}$

$\therefore \hat{f}(\omega) = Re\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(-a+ib+iw\right)|t|} \,dt \right\} = Re\left\{ \frac{1}{\sqrt{2\pi}} \frac{1}{-a+ib+iw} e^{-a|t|}.e^{i(b+\omega)|t|}|^\infty_{-\infty} \right\}$

I think I can argue that the $e^{-a|t|}$ term dominates the $e^{i}$ term which is bounded. BUT the lower limit will make
$e^{-a|t|}$ infinite... is there a better way of approaching this FT?

 

[Ackbach]

You should recall that
$$|t|=\begin{cases}\phantom{-}t,\; t\ge 0 \\ -t,\; t<0\end{cases}.$$
Then split up your integral into two pieces depending on where your integrand changes.

 

[ognik]

You should recall that
$$|t|=\begin{cases}\phantom{-}t,\; t\ge 0 \\ -t,\; t<0\end{cases}.$$

Thanks Ackbach and yes I should!

I'd appreciate if you'd please check the rest ... and sorry about the edits if we cross over.

What I did below seems wrong on reflection, I think I should take the real part only w.r.t. b - if that is possible?

If I did that, would the real part w.r.t. $\omega$ be the same as the Fourier Cosine Transform? And would the Fourier Sine Transform be just Imaginary part of that exponential Transform?

$$\hat{f}(\omega)= \frac{1}{\sqrt{2\pi}} Re\left\{ \int_{-\infty}^{0} e^{at}e^{eibt}e^{i\omega t} \,dt +
\int_{0}^{\infty} e^{at}e^{eibt}e^{i\omega t} \,dt \right\} $$

$$ =\frac{1}{\sqrt{2\pi}} Re\left\{ \frac{1}{a+i(b+\omega)} e^{t(a+i(b+\omega))} |^0_{-\infty} + \frac{1}{-a+i(b+\omega)} e^{t(-a+i(b+\omega))} |^\infty_0 \right\} $$

$$ = \frac{1}{\sqrt{2\pi}} Re\left\{ \frac{1}{a+i(b+\omega)} - \frac{1}{-a+i(b+\omega)} \right\}$$

$$= \frac{1}{\sqrt{2\pi}} \frac{2a}{a^2+(b+\omega)^2} ?$$

Also - would this be the same as the Fourier Cosine Transform? And would the Fourier Sine Transform be just Imaginary part of the exponential Transform?

 

[ognik]

Hi guys, what I would like to know is - can I do the FT using Euler's form for Cos(bt) - and then extract the real part w.r.t. b but leaving the complex parts w.r.t. wt? Or is the only possible approach here to use integration by parts?

 

[Ackbach]

I think your basic strategy is fine, but there are some details that may need improving:

\begin{align*}\hat{f}(\omega) &= \text{Re}\left\{ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\left(ib+iw\right)t} e^{-a|t|} \,dt \right\} \\
&=\frac{1}{\sqrt{2\pi}}\:\text{Re}\left\{ \int_{-\infty}^{0}e^{\left(ib+iw\right)t} e^{at} \,dt+
\int_{0}^{\infty}e^{\left(ib+iw\right)t} e^{-at} \,dt \right\} \\
&=\frac{1}{\sqrt{2\pi}}\:\text{Re}\left\{ \int_{-\infty}^{0}e^{\left(a+ib+iw\right)t} \,dt+
\int_{0}^{\infty}e^{\left(-a+ib+iw\right)t}\,dt \right\}.\end{align*}

Can you finish from here?