Final answer:
The family of functions δn(x) = sin(nx)/πx is shown to be a valid representation of the Dirac delta function by leveraging symmetry arguments to simplify integration, considering the properties of sinc functions, and demonstrating that integral contributions cancel symmetrically around the origin.
Step-by-step explanation:
The student's question pertains to demonstrating that the family of functions δn(x) = sin(nx)/πx is a valid representation of the Dirac delta function by evaluating the limit:
limn→∞ ∫ [∞] −[∞] f(x) sin(nx)/πx dx = f(0)
This can be shown using the properties of the sinc function and the Dirac delta function. The sinc function is defined as sin(x)/x, which appears as a scaling variant in our integral. Given that the Dirac delta function is spherically symmetric, we simplify the integration process by considering a symmetry about the origin, integrating from 0 to ∞ and doubling the result.
Here are the basic ideas to prove it:
- Use the fact that sin(nx)/πx is the nth Fourier sine coefficient of f(x) when considering the Fourier sine series expansion on the interval [0, ∞].
- Show that the integral over sin(nx)/πx converges to zero for values other than the origin as n goes to infinity, due to the oscillatory nature of the sine function which cancels contributions symmetrically on either side of the origin.
- Finally, demonstrate that as n approaches infinity, the integral of sin(nx)/πx with the function f(x) approaches f(0), as only the value at the origin has a non-canceled contribution in the limit.
Dimensional consistency and symmetry arguments are often helpful in similar proofs involving functions in physics, particularly wave functions. Notice also the importance of even and odd function properties, as neat cancellations result in simplifying integral evaluations in physics problems.