Why substitute $t = 0$ into the inverse Fourier transform to evaluate the Dirichlet integral?

Conditions

• The inverse Fourier transform formula is $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$.
• The function $f(t)$ is the rectangular pulse with $f(0)=1$.
• $F(\omega)$ is the previously computed Fourier transform $2 \frac{\sin \omega}{\omega}$.

Reasoning, step by step

1. Recall the inverse Fourier transform formula: $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$.
2. Choose $t = 0$ to eliminate the complex exponential term: $e^{i\omega \cdot 0} = e^0 = 1$.
3. Evaluate the left-hand side using the definition of the rectangular pulse: $f(0) = 1$.
4. Substitute the known expression for $F(\omega)$ on the right-hand side: $F(\omega) = 2 \frac{\sin \omega}{\omega}$.
5. Set up the equation: $1 = \frac{1}{2\pi} \int_{-\infty}^{\infty} 2 \frac{\sin \omega}{\omega} d\omega$.
6. Simplify the constants: $1 = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\sin \omega}{\omega} d\omega$.
7. Rearrange to solve for the full-axis integral: $\int_{-\infty}^{\infty} \frac{\sin \omega}{\omega} d\omega = \pi$.
8. Use the evenness of $\frac{\sin \omega}{\omega}$ to find the half-axis integral: $\int_{0}^{\infty} \frac{\sin \omega}{\omega} d\omega = \frac{\pi}{2}$.
9. Conclude that substituting $t=0$ is the key step that links the specific function value to the desired improper integral.

Example

The video states: 'The instructor chose to set $t = 0$ on both sides of the equation. At this point, the original function satisfies $f(0) = 1$, and the exponential term becomes $e^{i\omega \cdot 0} = 1$. Thus, the inverse transform formula reduces to $1 = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) d\omega$'.

Common misconceptions

• Thinking that $t=0$ is arbitrary; it is chosen specifically because $f(0)$ is known and non-zero, and it simplifies the kernel to 1.
• Assuming the integral converges absolutely at $t=0$ without considering the principal value or the specific properties of the sinc function; the video treats it as a standard improper integral evaluation.
• Confusing the forward and inverse transform formulas; the substitution is made in the inverse formula.

Watch the explanation

Explore next

• What is the value of $f(0)$ for the rectangular pulse?
• How does the evenness of $\sin (w)/w$ help in finding the final result?
• Why is the factor 1/(2pi) present in the inverse transform?