Error Function Calculator

Calculate the error function erf(x) with step-by-step solutions, inverse erf, and probability applications

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erf(x) = (2/√π) ∫₀ˣ exp(-t²) dt

Properties: erf(0) = 0 | erf(∞) = 1 | erf(-x) = -erf(x)

erf(x) ranges from -1 (at -∞) to 1 (at +∞) Valid range: -1 < y < 1

Quick Examples

Invalid Input

For erfinv(y), the input must be between -1 and 1 (exclusive).

erf() erfinv()

erfc() =

erfc(x)

Φ(x) Normal CDF

d/dx[erf(x)]

Step-by-Step Solution

1

Identify the input value

x = y =

2

Apply the error function formula Apply the inverse error function

erf(x) = (2/√π) ∫₀ˣ exp(-t²) dt Find x such that erf(x) = y

3

Result

erf() = erfinv() =

Common Values Reference

x erf(x) erfc(x)
-2 -0.9953 1.9953
-1 -0.8427 1.8427
0 0.0000 1.0000
0.5 0.5205 0.4795
1 0.8427 0.1573
1.5 0.9661 0.0339
2 0.9953 0.0047
3 0.9999978 0.0000022

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About Error Function Calculator

What is the Error Function?

The error function erf(x), also known as the Gauss error function, is a special function that appears frequently in probability, statistics, and partial differential equations.

Mathematical Definition

The error function is defined by the integral:

erf(x) = (2/√π) ∫₀ˣ exp(-t²) dt

This integral cannot be expressed in terms of elementary functions, which is why erf(x) is defined as a special function.

Key Properties

Fundamental Values

  • erf(0) = 0
  • erf(∞) = 1
  • erf(-∞) = -1

Odd Function Property

The error function is an odd function, meaning:

erf(-x) = -erf(x)

Derivative

The derivative of the error function is:

d/dx [erf(x)] = (2/√π) exp(-x²)

Taylor Series Expansion

erf(x) = (2/√π) × (x - x³/3 + x⁵/10 - x⁷/42 + ...)

erf(x) = (2/√π) × Σₙ₌₀^∞ [(-1)ⁿ × x^(2n+1)] / [n! × (2n+1)]

Relationship to Normal Distribution

For a standard normal distribution, the cumulative distribution function (CDF) is:

Φ(x) = ½[1 + erf(x/√2)]

The probability that a normally distributed variable falls within [-x, x] standard deviations is erf(x/√2).

Relationship to Complementary Error Function

erfc(x) = 1 - erf(x)

The complementary error function is useful for numerical precision when erf(x) is close to 1.

Common Values

x erf(x)
0 0.0000
0.5 0.5205
1.0 0.8427
1.5 0.9661
2.0 0.9953
2.5 0.9996

Applications

  1. Probability and Statistics - Normal distribution calculations
  2. Heat Transfer - Solutions to diffusion equations
  3. Signal Processing - Error probability in communications
  4. Physics - Particle diffusion, semiconductor physics
  5. Finance - Black-Scholes option pricing model

Inverse Error Function

The inverse error function erfinv(y) gives the value x such that erf(x) = y:

erfinv(erf(x)) = x

Valid for y ∈ (-1, 1).