Fourier Transform of a Step Function

Question

I'm trying to obtain the form of a sinc function that I know I'm suppose to get in Mathematica. I'm doing this because I intend to do a lot with Fourier Transforms and I'd like to know I'm not missing constants or something of the like.

$f(x) = A$ for $-a/2$ ≤ $x a/2$
$f(x) = 0$ otherwise

Employing the following:

FourierTransform[UnitStep[a/2 + x] UnitStep[a/2 - x], x, k, FourierParameters -> {1, -1}]

It gives: (2 Sin[(a k)/2] UnitStep[a])/k

Is this Mathematica's way of giving the sinc function?

I guess I was expecting something more like (2 Sin[(a k)/2])/(k a)

I know there are different parameters that can be used, but none of them have given the form I was expecting.

When I try to do the 'FT' directly as an integral:

Integrate[A Exp[-I k x], {k, -a/2, a/2}]

It gives: (2 A Sin[(a x)/2])/x

Are these the same?

I was expecting something that would lead to $\rm{sinc}[k a/2]$

I know the problem is me, but I have not found previous questions that can help, and the documentation in Mathematica (version 8) also has not helped either.

* Ok....thanks everyone. I appreciate it!

<><> Sorry I have not 'accepted' an answer. I was not aware that was to be done. I just found that out. Still learning my way around here. Thanks again to everyone's answer....very helpful.

Answer 1

In Mathematica there is a designated function for this, UnitBox,

PiecewiseExpand[UnitBox[x]]

which gives expected result without assumptions:

FourierTransform[UnitBox[x/a], x, k, FourierParameters -> {1, -1}]

Abs[a] Sinc[(a k)/2]

There is actually a set of designated functions:

FourierTransform[UnitTriangle[x/a], x, t, FourierParameters -> {1, 1}]

Abs[a] Sinc[(a t)/2]^2

FourierSinCoefficient[#[x], x, n, FourierParameters -> {1, 2 Pi}] & /@ 
{TriangleWave, SawtoothWave, SquareWave}

Answer 2

There is no inconsistency here. If you assume a to be positive, as Daniel mentioned, you get the same answer:

FourierTransform[UnitStep[a/2 + x] UnitStep[a/2 - x], x, k, 
    FourierParameters -> {1, -1}, Assumptions -> a > 0]
(* (2 Sin[(a k)/2])/k *)

Integrate[UnitStep[a/2 + x] UnitStep[a/2 - x] Exp[-I k x], 
    {x, -∞, ∞}, Assumptions -> a > 0]
(* (2 Sin[(a k)/2])/k *)

Note that:

$$ \frac{2\sin(a k/2)}{k} = \frac{a \sin (ak/2)}{ak/2} = a \text{sinc}(ak/2)$$