Gaussian positive-part expectation ($\mathbb{E}[(m+aZ{)}^{+}]=m\Phi (m/a)+a\varphi (m/a)$)

For $Z$ with standard normal distribution, $m\in \mathbb{R}$, and $a>0$, integrate $(m+az)\varphi (z)$ over $z>-m/a$. The identity ${\varphi }^{\prime }(z)=-z\varphi (z)$ gives the displayed expectation of the positive part. This is useful for option pricing when a financial payoff is the positive part of a normal random variable, including a log-call option. At $a=0$ the formula is interpreted by its limit, $m^{+}$.