for some holomorphic function g . But in order for F to belong to the Segal–Bargmann space —that is, for F to be square-integrable with respect to a Gaussian measure— g must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that g must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function.