How To Calculate Error Function

Error Function Formula:

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

Error Function Result:

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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 integral definition:

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

For practical computation, it uses 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, heat conduction problems, diffusion equations, and digital communications. It's essential in calculating bit error rates and confidence intervals.

4. Using the Calculator

Tips: Simply enter any real number x to calculate erf(x). The function is odd (erf(-x) = -erf(x)), so negative values are automatically handled.

5. Frequently Asked Questions (FAQ)

Q1: What's the range of the error function?
A: erf(x) ranges from -1 to 1, approaching ±1 as x approaches ±∞.

Q2: How accurate is this calculator?
A: The approximation used has maximum error of 1.5×10⁻⁷, sufficient for most practical applications.

Q3: 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)].

Q4: Are there related functions?
A: Yes, the complementary error function is erfc(x) = 1 - erf(x). There's also the imaginary error function erfi(x).

Q5: Can this be used for complex numbers?
A: This calculator handles real numbers only. Complex error function requires different computation methods.