Gaussian Error Function Calculator

Gaussian Error Function:

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

Error Function Value:

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1. What is the Gaussian 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 definition:

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

Where:

\( x \) — Input value (real number)
\( e \) — Base of natural logarithm (~2.71828)
\( \pi \) — Pi (~3.14159)

Explanation: The calculator implements a numerical approximation (Abramowitz and Stegun) that provides results accurate to 7 decimal places.

3. Applications of the Error Function

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

4. Using the Calculator

Tips: Enter any real number value for x. The calculator will return the value of erf(x) between -1 and 1. For large x (>3.5), erf(x) approaches ±1 very closely.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between erf and the normal distribution?
A: The cumulative distribution function (CDF) of a standard normal distribution can be expressed in terms of erf: Φ(x) = ½[1 + erf(x/√2)].

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

Q3: How accurate is this calculator?
A: The implementation uses an approximation with maximum error of 1.5×10⁻⁷, sufficient for most practical applications.

Q4: What is the complementary error function?
A: erfc(x) = 1 - erf(x), often used for large x values where erf(x) approaches 1.

Q5: Are there series expansions for erf?
A: Yes, for small x: erf(x) ≈ (2/√π)(x - x³/3 + x⁵/10 - x⁷/42 + ...).