How to get the integral result?

[Tspirit]

Homework Statement

I am studying Gerry's <Introductory Quantum Optics>, in which there is an integral (Eq. 4.37)
$$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\frac{\pi}{2}t.$$
I don't know how to get the result of the right side.

Homework Equations

I have no idea.

The Attempt at a Solution

I tried like the following
$$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\intop_{-infinity}^{+infinity}\frac{1-cos(\triangle t)}{2\triangle^{2}}d\triangle$$
But don't know what to do next.

 

[BvU]

Matter of googling "integral sin^2x/x^2" to see our math colleagues have a thread on this one...

But for more specific questions PF is of course available with a

 

[Ray Vickson]

Homework Statement

I am studying Gerry's <Introductory Quantum Optics>, in which there is an integral (Eq. 4.37)
$$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\frac{\pi}{2}t.$$
I don't know how to get the result of the right side.

Homework Equations

I have no idea.

The Attempt at a Solution

I tried like the following
$$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\intop_{-infinity}^{+infinity}\frac{1-cos(\triangle t)}{2\triangle^{2}}d\triangle$$
But don't know what to do next.

In questions of this type you should always try to simplify first, before doing anything else. So if we use the notation $w$ instead of your $\Delta$ and $a$ instead of your $t/2$, we have an integral of the form
$$I(a) = \int_{-\infty}^{\infty} \frac{\sin^2(a w)}{w^2} \, dw.$$
Change variables to $u = a w$ to get $I(a) = a K$, where
$$K= \int_{-\infty}^{\infty} \frac{\sin^2 u}{u^2} \, du.$$
Now $K$ is just some constant; it happens to equal $\pi$, but that is another issue.

 

[hunt_mat]

I think there is a simple solution. Take:

$$I(a)=\int_{-\infty}^{\infty}\frac{\sin^{2}ax}{x^{2}}dx$$

Now differentiate w.r.t. $a$ to find that:

$$I'(a)=2\int_{-\infty}^{\infty}\frac{\sin 2ax}{2x}dx$$

Now the integral on the RHS can be calculated using residue calculus. It's result will be a function of $a$, the result is found out via integration.