Question

\text { 3.2.13 The step function } s(x) \text { is zero for } x \leq 0 \text { and jumps to one for } x>0 \text { ; its

derivative is the } \text { delta function with } \delta(x)=0 \text { for every } x \neq 0 \text { but } \int \delta(x) d x=1 \text { . To make this reasonable, } \text { consider the step } s \text { as a limit of functions } s_{n} \text { which have slope } n \text { between } x=0 \text { and } x=1 / n \text { elsewhere } s_{n}^{\prime}=0 \text { . Compute } \int s_{n}^{\prime} d x \text { and } \int\left(s_{n}^{\prime}\right)^{2} d x \text { , and as } n \rightarrow \infty \text { verify formally that } \int \delta=1 \text { and } ^^20\int ^^20\delta^2=\infty^^20.^^20

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