Erfð

erfð is not a standard mathematical term. It is most likely a typographical variation or misprint of erf, the error function, which is a well established special function in probability, statistics, and analysis.

The error function erf is defined by erf(x) = (2/√π) ∫ from 0 to x of e^(−t^2) dt. It is an odd, monotone-increasing function with erf(0) = 0, and its range is (−1, 1). The derivative satisfies erf′(x) = (2/√π) e^(−x^2). The complementary error function erfc(x) = 1 − erf(x) is erfc(x) = (2/√π) ∫ from x to ∞ e^(−t^2) dt, and erfc is often used for tail probabilities.

A frequent relationship is with the standard normal distribution: the CDF Φ(z) = (1/2)[1 + erf(z/√2)]. This connects the error function to Gaussian probabilities. Series expansions around x = 0 give erf(x) = (2/√π)[x − x^3/3 + x^5/10 − ...], while asymptotic forms for large x describe the behavior of erfc(x), e.g., erfc(x) ~ (e^(−x^2) / (√π x)) [1 − 1/(2x^2) + 3/(4x^4) − ...].

Numerically, erf is implemented in many mathematical libraries and software environments, with efficient algorithms for accurate evaluation across the real line. In texts where "erfð" appears, it should be treated as a likely typographical error or nonstandard notation for erf, rather than a separate mathematical object. The core concepts and properties above apply to the conventional erf.