This problem rebuilds the composition rule for one function

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

This problem rebuilds the composition rule for one function, starting from the scaling rule δ(ax)=δ(x)/∣a∣\delta(ax)=\delta(x)/|a| alone, and then uses it. ((a)) Let g(x)=x3−4xg(x) = x^3 - 4x. Find the zeros of gg and confirm each is simple by checking g′(xi)≠0g'(x_i)\neq0 at each. ((b)) Find the weight with which each zero contributes to δ(g(x))\delta(g(x)), justifying it from the scaling rule, and write δ(x3−4x)\delta(x^3-4x) as a sum of three weighted spikes. ((c)) Use the result to evaluate ∫−∞∞e−x2 δ(x3−4x) dx\int_{-\infty}^{\infty} e^{-x^2}\,\delta(x^3-4x)\,\mathrm{d}x, to three decimal places.

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