The $\operatorname{erf}$ function

Can a function be 'defined' as the anti-derivative of another function?

$\operatorname{erf}$ is one such function. Even though it has various proper infinite series expansions (as in Wolfram MathWorld - Erf), it is defined by mathematicians as such:
$$\operatorname{erf}(z) \equiv \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$
This function has extensive implications in statistics, and can be used to express the integral of $e^{-x^2}$ and $e^{x^2}$. See the post that tries to find integral of $e^{x^2}$ - we need not use integration by parts there, but rather can use the $\operatorname{erf}$ function.

Source: Wolfram MathWorld

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