EDB — 1DM

Exercises

1. [1DM]Prerequisites:[1DJ]↺↻,[09N]↺↻.Show that the function

   \begin{equation} 𝜑(x) = \begin{cases} e^{-1/x} & \text{if}~ ~ x>0 \\ 0 & \text{if}~ ~ x≤ 0 \end{cases} \label{eq:Cinfty_ non_ analitica} \end{equation}

   20

   is of class \(C^∞\), and for \(x{\gt}0\)

   \begin{align*} 𝜑^{(n)}(x)= e^{-1/x} ∑_{m=1}^ n \binom {n-1}{m-1} \frac{n! }{ m! } \frac{(-1)^{m+n}}{x^{m+n}}\quad , \\ \quad \binom {n-1}{m-1}= \frac{(n-1)!}{(n-m)!(m-1)!}\quad . \end{align*}

   whereas \(𝜑^{(n)}(x)=0\) for each \(n∈ℕ,x≤ 0\).

   Proceed similarly to

   \begin{equation} 𝜓(x) = \begin{cases} e^{-1/|x|} & \text{if}~ ~ ~ x≠ 0 \\ 0 & \text{if}~ ~ x= 0 \end{cases} \label{eq:Cinfty_ non_ analitica_ bis} \end{equation}

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   again \(𝜓∈ C^∞\) and \(𝜓^{(n)}(0)=0\) for each \(n∈ℕ\); but in this case \(𝜓(x)=0\iff x=0\).

   Solution 1

   [1DN]↺↻