Error Function Calculator

What is the Error Function?

The error function, denoted as erf(x), is a special function of sigmoid shape that occurs in probability, statistics, and partial differential equations. It is defined as:

\[erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt\]

Where:

• \(x\) is the input value
• \(e\) is Euler's number (approximately 2.71828)
• \(\pi\) is pi (approximately 3.14159)

Properties of the Error Function

• erf(0) = 0
• erf(-x) = -erf(x) (odd function)
• erf(∞) = 1
• erf(-∞) = -1
• The function has a sigmoid shape

Calculation Steps

1. Input the value of x
2. Use an approximation method (as the integral doesn't have a closed-form solution)
3. Apply the approximation formula
4. Calculate the result

Example Calculation

Let's calculate erf(1):

1. Input: x = 1
2. Using the approximation method (details omitted due to complexity)
3. Result: erf(1) ≈ 0.8427

Visual Representation

This graph shows the characteristic S-shaped curve of the error function. The function is odd (symmetric about the origin), and its values range from -1 to 1 as x goes from negative infinity to positive infinity.