Error Function Calculator
Calculate the error function erf(x) with step-by-step solutions, inverse erf, and probability applications
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
Identify the input value
x = y =
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
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
- Probability and Statistics - Normal distribution calculations
- Heat Transfer - Solutions to diffusion equations
- Signal Processing - Error probability in communications
- Physics - Particle diffusion, semiconductor physics
- 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).