2.5.3 So­lu­tion eigvals-c

Ques­tion:

Show that $\sin(kx)$ and $\cos(kx)$, with $k$ a con­stant, are eigen­func­tions of the in­ver­sion op­er­a­tor ${\mit\Pi}$, which turns any func­tion $f(x)$ into $f(-x)$, and find the eigen­val­ues.

An­swer:

By de­f­i­n­i­tion of ${\mit\Pi}$, and then us­ing [1, p. 43]:

$${\mit\Pi} \sin(kx) = \sin(-kx) = - \sin(kx) \qquad{\mit\Pi} \cos(kx) = \cos(-kx) = \cos(kx)$$

So by de­f­i­n­i­tion, both are eigen­func­tions, and with eigen­val­ues $\vphantom{0}\raisebox{1.5pt}{$-$}$1 and 1, re­spec­tively.