Someone argues: " is defined as the limit of a rectangle of height and width as

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Delta functions, distributions and generalised functions

Problem

Someone argues: "δ(x)\delta(x) is defined as the limit of a rectangle of height 1/b1/b and width bb as b→0b\to0. At b=0.010b = 0.010 the height is already 100100, and it keeps growing without bound as bb shrinks further. So δ(0)=∞\delta(0) = \infty, and that infinity is the actual value of the delta function at the origin." Now take a different family: sb(x)s_b(x) equal to 1/b1/b for b/2<∣x∣<bb/2<|x|<b and zero elsewhere, a rectangle of width 2b2b with its middle removed. ((a)) Show that sbs_b has unit area for every bb, and evaluate ∫−∞∞sb(x)cos⁡x dx\int_{-\infty}^{\infty}s_b(x)\cos x\,\mathrm{d}x at b=0.10b=0.10 and at b=0.010b=0.010. ((b)) Explain why ∫−∞∞sb(x)f(x) dx→f(0)\int_{-\infty}^{\infty}s_b(x)f(x)\,\mathrm{d}x\to f(0) as b→0b\to0 for every continuous ff, and give sb(0)s_b(0). ((c)) Explain what is wrong with the claim, and state what, if anything, can correctly be asserted about δ(0)\delta(0).

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