# Good choice of notation for normal distribution pdf/cdf

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?

$$\Phi_{\mu, \sigma} (x)\equiv \int_{-\infty}^x\frac{\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right)}{\sqrt{2\pi\sigma^2}}dy$$
and then use the left-hand side symbol. This up to a degree aligns with the convention that $\Phi_2(x,y;\rho)$ denotes the bivariate standard normal cdf.
Then the univariate case could be written $\Phi_1(x)$, to which we can arrive by starting from $\Phi_{\mu, \sigma} (x)$, then setting $\mu =0, \sigma=1$, and eliminate the zero-subscript (since absence many times essentially means "zero"). This is essentially a happy case of confounding.