Fourier transform of rectangular pulse (Waves)

[tigger88]

Homework Statement

F(w) is the Fourier transform of f(t). Write down the equation for F(w) in terms of f(t).
A rectangular pulse has height H and total length t0 in time. Show that as a function of w, the amplitude density is propertional to sinc(wt0/2).

Homework Equations

F(w) = integral from -infinity to +infinity of: f(t)exp(-iwt)dw

The Attempt at a Solution

integral from -t0/2 to +t0/2 of: h*exp(-iwt)dw

I have access to the solution to this problem, which says that it should be:
integral from -t0/2 to +t0/2 of: h*exp(-iwt)dt,
but I don't understand why I'm integrating wrt t now, when the definition says w.

Could somebody please explain this?

Thanks in advance.

 

[Cyosis]

Where do you get that definition from? Think about it, you want to find a function $F(\omega)$, but if you calculate the integral you've written down as the "definition" then the integration boundaries will be inserted into $\omega$ after the integration. As as a result you won't have a function with variable $\omega$.

The correct definition is (normalization conventions can be different):
$$F(\omega)}=\int_{-\infty}^\infty e^{-i \omega t} dt$$

 

[tigger88]

Actually I got that definition from the solution to the question. It makes a whole lot more sense now, thanks for your reply!