Consider the eigenvalue problem

$\frac{d^2()}{d{}^2}=-m^2()$

where $m$ is a real $($not imaginary nor complex$)$ number. The two eigenfunctions of hat$(A)=\frac{d^2}{d}{}^2$ are

${}_m()=e^{im},$ and $,{}_{-m}()=e^{-im}$

We can easily show that each of these eigenfunctions has the eigenvalue $-m^2.$ Show that any linear combination of ${}_m()$ and ${}_{-m}()$ is also an eigenfunction of hat$(A)=\frac{d^2}{d}{}^2.$