The Gaussian family

PHYSICS · Subjective · 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

The Gaussian family gσ(x)=1σ2π e−x2/2σ2g_\sigma(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-x^2/2\sigma^2}, of unit area for every σ>0\sigma>0, is one of the delta sequences: ∫−∞∞f(x) gσ(x) dx→f(0)\int_{-\infty}^{\infty} f(x)\,g_\sigma(x)\,\mathrm{d}x \to f(0) as σ→0\sigma\to0 for every smooth ff. (a) Differentiate gσ(x)g_\sigma(x) with respect to xx and show that gσ′(x)=−xσ2gσ(x)g_\sigma'(x) = -\dfrac{x}{\sigma^2}g_\sigma(x), and that gσ(x)→0g_\sigma(x)\to0 as x→±∞x\to\pm\infty for every fixed σ\sigma. (b) For a smooth function f(x)f(x) with ff and f′f' bounded, integrate ∫−∞∞f(x) gσ′(x) dx\int_{-\infty}^{\infty} f(x)\,g_\sigma'(x)\,\mathrm{d}x by parts and use the sifting property of gσg_\sigma stated above to show that this integral tends to −f′(0)-f'(0) as σ→0\sigma\to0. (c) State the resulting rule for ∫−∞∞f(x) δ′(x−a) dx\int_{-\infty}^{\infty} f(x)\,\delta'(x-a)\,\mathrm{d}x and use it to evaluate ∫−∞∞(3x2+2x) δ′(x−1) dx\int_{-\infty}^{\infty}(3x^2+2x)\,\delta'(x-1)\,\mathrm{d}x.

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