Error Function Calculator (erf / erfc)

Calculate the error function erf(x) and complementary error function erfc(x) with series expansion details.

Select Function & Enter Value

Result

erf(1)
0.842701
error function value
erf(x)0.842701
erfc(x) = 1 - erf(x)0.157299
Input x1
Series terms used10

Step-by-Step Solution

erf(1) = (2/sqrt(pi)) * sum = 0.842701

Understanding the Error Function

The error function, commonly denoted as erf(x), is a special function of sigmoid shape that occurs in probability, statistics, and partial differential equations. It is defined as the integral of the Gaussian (normal) distribution and is closely related to the cumulative distribution function of the normal distribution.

The error function is defined as: erf(x) = (2/sqrt(pi)) * integral from 0 to x of e^(-t^2) dt. The complementary error function is erfc(x) = 1 - erf(x).

Key Properties and Formulas

Taylor Series

The error function can be computed via its Maclaurin series expansion.

erf(x) = (2/sqrt(pi)) * SUM(-1)^n * x^(2n+1) / (n! * (2n+1))

Complementary Error Function

erfc(x) measures the probability in the tail of the normal distribution.

erfc(x) = 1 - erf(x)

Symmetry Property

The error function is an odd function, symmetric about the origin.

erf(-x) = -erf(x)

Limiting Values

The error function approaches +/-1 as x approaches +/-infinity.

erf(0) = 0, erf(inf) = 1

Normal Distribution Link

The CDF of the standard normal distribution can be expressed using erf.

Phi(x) = (1/2)[1 + erf(x/sqrt(2))]

Derivative

The derivative of erf(x) is a Gaussian function.

d/dx erf(x) = (2/sqrt(pi)) * e^(-x^2)

Applications of the Error Function

The error function appears throughout science and engineering. In statistics, it defines confidence intervals and p-values. In physics, it describes heat diffusion and particle diffusion. In signal processing, it is used for Gaussian noise analysis. In finance, it relates to the Black-Scholes model through the cumulative normal distribution.

Common Values

  • erf(0) = 0
  • erf(0.5) = 0.520500 (approximately)
  • erf(1.0) = 0.842701 (approximately)
  • erf(1.5) = 0.966105 (approximately)
  • erf(2.0) = 0.995322 (approximately)
  • erf(3.0) = 0.999978 (approximately)