AngouriMath

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Erf

Method with 2 overloads

Erf(AngouriMath.Entity.Number.Complex)
Summary
The error function, erf(z) = 2/sqrt(pi) int_0^z e^(-t^2) dt, at the working
precision, for every complex z.

Remarks
With P digits of working precision, up to |z|^2 = P ln 10 it is summed as
erf(z) = 2/sqrt(pi) e^(-z^2) sum 2^n z^(2n + 1)/(1 3 5 ... (2n + 1)) (Abramowitz and Stegun 7.1.6). The terms are all positive on the real line, so a
real argument loses nothing to cancellation. Off it they turn, and the sum loses up
to 2 Im(z)^2/ln 10 digits, which are carried as guard digits.
Beyond that it is 1 - erfc(z), with erfc(z) from its asymptotic series
e^(-z^2)/(z sqrt(pi)) sum (-1)^n (2n - 1)!!/(2 z^2)^n cut at its smallest
term. That term is below e^(-|z|^2), and so below the precision there. An
argument in the left half-plane is folded over by erf(-z) = -erf(z) first,
since the asymptotic series holds for |arg z| < 3 pi/4.
The result is rounded to DecimalPrecisionContext, whose
exponent range bounds what can be represented: erfc(30) is about
2.6e-393 and comes out as zero there, and erfi(50), about
e^2500, as infinity. The working contexts have no such bound.
https://github.com/asc-community/AngouriMath/issues/1501
Erf(PeterO.Numbers.EDecimal,PeterO.Numbers.EDecimal,PeterO.Numbers.EContext,System.Int32)
Summary
erf(re + i im) in a context of context's precision and
extraDigits more, before any rounding back.