Error Function Calculator
Calculate error function (erf), complementary error function (erfc), and their inverses
Error Function Calculator
Error Function
Result
Example Values
x | erf(x) |
---|---|
0 | 0.000000 |
0.5 | 0.520500 |
1 | 0.842701 |
1.5 | 0.966105 |
2 | 0.995322 |
2.5 | 0.999593 |
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
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