On Wikipedia I found the definition of the error function $$\text{erf}(z)=\frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2} \, \text{d}t$$ for a normally distributed random variable $X$ with mean 0 and standard deviation $\frac{1}{\sqrt{2}}$. Then erf$(z)$ then gives the probability that $X$ falls in the range $[-x,x]$.
Wikipedia also says, that the complementary error function (erfc), defined as $$ \text{erfc}(z)=1-\text{erf}(z),$$ has an upper bound given by $$\text{erfc}(z) \leq e^{-z^2}. $$
Now I asked myself if one can also define an error function for a different standard deviation?
For $\sigma =\frac{\sqrt{k}e}{v}$ being my standard deviation (with $k,e,v>0$ constants), is it possible to define $$\text{Erf}(k,v,e,z)= \frac{v}{\sqrt{2\pi k}\cdot e}\int_0^z \exp\left(\frac{-x^2v^2}{2ke^2}\right)\, \text{d}x ?$$ With this new error function, could I also find an upper bound for the complementary error function?
The error function $\operatorname{erf}(x)$ is a specific function and is not equipped with parameters. That is to say, there is only one function we mean by the expression $\operatorname{erf}(x)$, in the same way that $x^2$ and $\sqrt{x}$ represent specific, unique functions. And likewise, we can say that $\operatorname{erf}(2)=0.995\ldots$, and not some other value, just as $\sqrt{2}$ equals the particular value of $1.414\ldots$
However, $\operatorname{erf}(x)$ is intimately tied to $\Phi(x |\, \mu, \sigma^2)$, the family of cumulative distribution functions of normal distributions, with mean $\mu$ and variance $\sigma^2$. Every cumulative normal distribution can be regarded as some particular transformation (scaling and translation) of $\operatorname{erf}(x)$, according to the parameters $\mu, \sigma^2$. In particular, we have
$$\Phi(x |\, \mu, \sigma^2) = \frac12\left[1+\operatorname{erf}\left(\frac1{\sqrt{2}}\frac{x-\mu}\sigma\right)\right]$$
So, $\operatorname{erf}(x)$ can be considered as the standard function that represents the family, in the same way as how all parabolas and lines in the plane can be considered as transformations of the functions $x^2$ and $x$.