Erf Calculator

Error Function:

\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \]

Error Function Value:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Error Function?

The error function (erf) is a special function in mathematics that describes the probability of a random variable falling within a certain range in a normal distribution. It's widely used in statistics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the following mathematical definition:

\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \]

Where:

\( x \) — Input value (unitless)
\( e \) — Euler's number (~2.71828)
\( \pi \) — Pi (~3.14159)

Explanation: The calculator implements a numerical approximation (Abramowitz and Stegun approximation) with maximum error of 1.5×10⁻⁷.

3. Applications of Error Function

Details: The error function is used in probability theory, heat conduction problems, diffusion equations, and digital communications. It's essential in calculating normal distribution probabilities.

4. Using the Calculator

Tips: Enter any real number (positive or negative). The result will be between -1 and 1, with erf(0) = 0 and erf(∞) = 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between erf and erfc?
A: erfc(x) = 1 - erf(x) is the complementary error function, often used for large x values.

Q2: What are the boundary values of erf?
A: erf(0) = 0, erf(∞) = 1, erf(-∞) = -1, and erf(-x) = -erf(x).

Q3: How accurate is this calculator?
A: It uses a polynomial approximation with maximum error of 1.5×10⁻⁷.

Q4: Can I calculate inverse erf with this?
A: No, this calculator only computes erf(x). Inverse erf requires different methods.

Q5: Why is erf important in statistics?
A: It gives the probability a normal random variable falls within [-x, x] standard deviations from the mean.