  After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau
threefolds, which we call the BCOV invariant and which we obtain using analytic
torsion. We give an explicit formula for the BCOV invariant as a function on
the compactified moduli space, when it is isomorphic to a projective line. As a
corollary, we prove the formula for the BCOV invariant of quintic mirror
threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa.
