Error Function Calculator

Calculate error function (erf), complementary error function (erfc), and their inverses

Error Function Calculator

Error Function

Formula: erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt
Domain: All real numbers
Range: (-1, 1)
Description: The Gaussian error function, fundamental in probability and statistics

Result

0.8427007929
erf(x)
Result
Input
1
Function
erf(x)
Status
Valid

Example Values

xerf(x)
00.000000
0.50.520500
10.842701
1.50.966105
20.995322
2.50.999593

Quick Reference

erf(x)

Domain: ℝ, Range: (-1, 1)

Probability within [-x, x]

erfc(x)

Domain: ℝ, Range: (0, 2)

Tail probability

erf⁻¹(x)

Domain: (-1, 1), Range: ℝ

Inverse of erf

erfc⁻¹(x)

Domain: (0, 2), Range: ℝ

Inverse of erfc

Properties

erf(-x) = -erf(x) (odd function)
erfc(x) = 1 - erf(x)
erf(0) = 0, erfc(0) = 1
lim(x→∞) erf(x) = 1
lim(x→∞) erfc(x) = 0

Understanding the Error Function

Mathematical Definition

The error function erf(x) is defined as the integral of the Gaussian distribution from 0 to x, scaled by 2/√π. It's fundamental in probability theory, statistics, and mathematical physics, particularly in solutions to the heat equation and normal distribution calculations.

Applications

  • Probability calculations with normal distributions
  • Heat transfer and diffusion equations
  • Statistical hypothesis testing
  • Signal processing and communications

Function Relationships

Error Function

erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt

Complementary

erfc(x) = 1 - erf(x)

Inverse Functions

Useful for finding quantiles and confidence intervals

Computational Methods

Series Expansion

Taylor series for accurate computation

Approximations

Various mathematical approximations available

Newton-Raphson

Iterative method for inverse functions