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The function $g(\omega) = \frac{\sin \omega}{\omega}$ is an even function because $\sin(-\omega) = -\sin(\omega)$ and $-\omega$ in the denominator cancels the negative sign, resulting in $g(-\omega) = g(\omega)$. For any even function, the integral over the symmetric interval $(-\infty, \infty)$ is equal to twice the integral over the half-interval $(0, \infty)$.

Conditions: The integrand is $\frac{\sin \omega}{\omega}$.; The integration interval is symmetric about the origin.; The value of the full-axis integral is known to be $\pi$.

The function $g(\omega) = \frac{\sin \omega}{\omega}$ is an even function because $\sin(-\omega) = -\sin(\omega)$ and $-\omega$ in the denominator cancels the negative sign, resulting in $g(-\omega) = g(\omega)$. For any even function, the integral over the symmetric interval $(-\infty, \infty)$ is equal to twice the integral over the half-interval $(0, \infty)$.

Conditions: The integrand is $\frac{\sin \omega}{\omega}$.; The integration interval is symmetric about the origin.; The value of the full-axis integral is known to be $\pi$.

The Fourier transform of the rectangular pulse function $f(t)$ is computed by substituting its piecewise definition into the integral definition of the Fourier transform. Because $f(t)$ is zero outside the interval $[-1, 1]$ and equals 1 inside, the infinite integral reduces to a finite integral over $[-1, 1]$.

Conditions: The function is defined as $f(t) = 1$ for $|t| \le 1$ and $0$ otherwise.; The Fourier transform is defined as $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$.; $\omega$ is a real number.

The Fourier transform of the rectangular pulse function $f(t)$ is computed by substituting its piecewise definition into the integral definition of the Fourier transform. Because $f(t)$ is zero outside the interval $[-1, 1]$ and equals 1 inside, the infinite integral reduces to a finite integral over $[-1, 1]$.

Conditions: The function is defined as $f(t) = 1$ for $|t| \le 1$ and $0$ otherwise.; The Fourier transform is defined as $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$.; $\omega$ is a real number.