202105241144 Homework 1 (Q4)

$\displaystyle{\int \delta (f(x))g(x)\,\mathrm{d}x}=\, ?$

Attempts.

$\begin{aligned} &\quad \int \delta(f(x))g(x)\,\mathrm{d}x \\ & = \int\delta (f(x))g(f^{-1}f(x))\frac{\mathrm{d}f(x)}{f'(x)} \\ & = \frac{g(f^{-1}f(x))}{f'(x)}\bigg|_{f(x)=0} \\ & = \frac{g(f^{-1}(0))}{f'(f^{-1}(0))} \end{aligned}$

Solution.

(The solution below is based on the manuscript of 2016-2017 PHYS3450 Electromagnetism Homework 1 Solution.)

Let the roots to $f(x)=0$ be $a_i$ where $i=1,2,3,\dots$ .
From $y=f(x)$ (or, $x=f^{-1}(y)$ ), we have $\mathrm{d}y=f'(x)\,\mathrm{d}x$ .

$\begin{aligned} &\quad \int \delta (f(x))g(x)\,\mathrm{d}x \\ & = \sum_i \int\delta (f(x))g(x)\,\mathrm{d}x\\ & = \sum_i \int\frac{\delta (y)g(f^{-1}(y))}{f'(f^{-1}(y))}\,\mathrm{d}y \\ & = \sum_i \frac{g(f^{-1}(0))}{f'(f^{-1}(0))} \\ & = \sum_i \frac{g(a_i)}{f'(a_i)} \\ \end{aligned}$

QED

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