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 error function formula:

Where:

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

Explanation: The function calculates the integral of the Gaussian function from 0 to x, normalized by 2/√π.

3. Applications of Error Function

Details: The error function is used in probability theory, heat conduction problems, diffusion equations, and digital communications (bit error rates).

4. Using the Calculator

Tips: Enter any real number x to calculate its error function value. The result will be between -1 and 1.

5. Frequently Asked Questions (FAQ)

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

Q2: What are the values at special points?
A: erf(0) = 0, erf(∞) = 1, erf(-∞) = -1. The function is odd: erf(-x) = -erf(x).

Q3: How is this different from complementary error function?
A: erfc(x) = 1 - erf(x). They're complementary functions.

Q4: Is there an inverse error function?
A: Yes, erf⁻¹(x) exists since erf is strictly increasing.

Q5: What's the Taylor series expansion?
A: erf(x) = (2/√π)(x - x³/3 + x⁵/10 - x⁷/42 + ...) for |x| < ∞.