EDB — 19M

[19M] Let \(p\) be a polynomial (with complex coefficients); fix \(𝜃∈ℂ, 𝜃≠ 0\). Define \(f(x)=-∫_ 0^ x e^{-𝜃 t} p(t)\, {\mathbb {d}}t\). Show that \(f(x)=e^{-𝜃 x}q(x)-q(0)\) where \(q\) is a polynomial that has the same degree as \(p\). Determine the linear map (i.e. the matrix) that transforms the coefficients of \(p\) into the coefficients of \(q\); and its inverse.