In this problem we outline a proof that the eigenfunctions of the Sturm-Liouville problem (1), (2) are real-valued. a. Let be an eigenvalue and let ( x) = U( x) + iV ( x) be a corresponding eigenfunction. Show that U and V are also eigenfunctions corresponding to . b. Using Theorem 11.2.3, or the result of 20, show that U and V are linearly dependent. c. Show that must be real-valued, apart from an arbitrary multiplicative constant that may be complex.

Week2BIOB170Notes Prokaryotes Bacteria • Mostwidespreadformsoflife • Ecologicalniches • Metabolicdiversity • Distinguishingtraits o Flagellaapparatuswithrotatoryfunction § Stiffandtubular § Poweredbybasalapparatus...