The error function got its name from the mathematics of measurement mistakes
In 1871 the mathematician J. W. L. Glaisher named a stubborn integral the error function, because it answered a practical question from the theory of errors: how likely is a measurement to land within a given distance of the truth? The name stuck, and so did the function.
The error function, often written erf, comes from integrating the bell-shaped curve that describes normally distributed quantities. Plotted, it traces a smooth S-shaped curve of the kind called a sigmoid. When a series of measurements scatters around the true value according to a normal distribution, erf tells you the probability that any single reading's error falls inside a chosen band. Glaisher used it to compute exactly that, the chance of an error lying between two limits, and he discussed its partner, the complementary error function, which is simply one minus erf, in another paper the same year.
Its uses have spread well beyond laboratory tables. Engineers use it to estimate the bit error rate of digital communication links, the chance that noise flips a transmitted bit. Both erf and its complement appear in solutions of the heat equation, describing how warmth spreads when a boundary suddenly jumps from one temperature to another. Statisticians lean on it and its approximations to bound the odds of rare events far out in the tails of a distribution.
Mathematically it is a curious object. The integral behind it cannot be written in closed form using elementary functions, a fact that follows from Liouville's theorem, so it has to be computed by other means, for example by expanding it as an infinite series and integrating term by term. It is an odd function, meaning that flipping the sign of the input flips the sign of the output, because the bell curve it integrates is symmetric. It is also an entire function with no singularities except at infinity, so its Taylor series always converges, although for large inputs the terms cancel so badly that the series becomes impractical.
Some older texts define it without the leading scaling factor, a reminder to check conventions before trusting a formula.
Source: Error function
