It is unfortunate that changes from Glaisher’s original $\mathrm {Erf}$: the switch of limits, names and the standardizing factor, did not apply to what Glaisher acknowledged was its most important …

The naïve (alternating) Maclaurin series is not really that numerically sound; I had already mentioned in my answer the modified series that has much better properties for computing $\mathrm {erf} (x)$ …

Well, there's a definition of erf and a definition of the Normal CDF.. The relations, derivable by some routine calculations, are shown as to how to convert between them, and how to convert between …

Aug 15, 2016 · I consider $\tanh { (x)}$ to be the somewhat prettyier function, and so I wanted to find an approximation to $\text {erf}$ with "nice functions" by a short expression.

Aug 27, 2014 · Looking at the graph of $\mathrm {erf}$ we find that it is symmetric about the origin, so approximations can be restricted to the positive half-plane. The graph further suggest two basic …

The inverse functions are quantile functions for Incomplete Gamma function based distributions and $\erf ^ {-1}$ is the quantile function for the Normal Distribution:

May 10, 2018 · The value of the integral starts at zero and grows toward the asymptote one. It reaches one only when z reaches infinity (so to speak, b/c/ infinity is not a number).

1 Context for using the Taylor Series of $$e^ {-t^2}$$ to find the Taylor expansion of the ERF function is found at Robert Ghrist/UPenn's Calculus wiki.

As you suspect, Taylor series expansions are only accurate near one point. To obtain a polynomial approximation that's good over an entire interval, the standard technique is to use Chebyshev …

Nov 26, 2021 · You can obtain a result for the second integral in terms of Owen's T-function which is defined as $$ T (x, p) = \frac {1} {2\pi}\int_ {0}^ {p} \frac {e^ {-\frac {1} {2} x^2 (1+t^2)}} {1+t^2} \, dt …