Let $\Phi(x)$ be the cumulative distribution function for the standard normal distribution. I have an expression with several terms of the form
$$\Phi\left(\frac{\mu-g(x)}{\sqrt{\sigma^2}}\right)$$ and $$\Phi\left(\frac{\mu+h(x)}{\sqrt{\sigma^2}}\right)$$
where both the expressions for $\mu$, $g(x)$ and $h(x)$ are fairly complicated.
Is there a different way to write these so that the expressions are not so complicated? What I would like is to e.g. write
$$\Phi\left(-g(x);\mu,\sigma^2\right) $$ and $$\Phi\left(h(x);\mu,\sigma^2\right) $$ but I am not sure that it's in good style, since the $\Phi$ denotes the standard normal distribution.
Another way to do it is of course to use
$$F(x;\mu,\sigma^2)=\int_{-\infty}^x\frac{\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right)}{\sqrt{2\pi\sigma^2}}dy$$
but I guess my question is, does there exist some convention for this?