Error Function Calculator Erf(x)

Error Function Calculator erf(x)
x =

Form accepts both decimals and fractions.

The error function, denoted erf, is defined by the integral

erf(x) = (2/√π)xo e-t2 dt.

Erf(x) is closely related to the normal probability curve; the cumulative distribution function of a normally distributed random variable X is

CDF(X) = 0.5 + 0.5erf[(X-μ)/σ√2],

where μ is the mean and σ is the standard deviation of the distribution. The error function integral cannot be evaluated in terms of elementary function, so one must use numerical algorithms. The error function is an odd function whose limit is -1 for negative values of x, and 1 for positive values of x. The function rapidly converges to its asymptotic values; erf(3) = 0.99998 and erf(-3) = -0.99998.

graph of erf(x)

Properties and Equations

The values of x for which x = erf(x) are approximately 0.6175 and -0.6175. If you don't have access to an error function calculator such as the one above, you can approximate the function with the formula
erf(x) approximation formula
The error function can also be expressed with infinite series and continued fractions:
erf(x) series and continued fraction representation
You can approximate the inverse of the error function, erf-1(x), by inverting the approximation formula above. The inverse of erf has several interesting derivative and integral properties.
expressions involving the inverse error formula, approximation for the inverse of erf

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