Complex K-theory of odd-dimensional real projective space ($K^{\ast }({\mathbb{RP}}^{2n+1})$)

If $L$ is the complexification of a real vector bundle of the tautological bundle and $\mu =[L]-1$, thenRealizing Real projective space as the circle bundle of the square of the tautological bundle on Complex projective space gives this through the K-theory Gysin sequence of a sphere bundle.