hibafüggvény

A hibafüggvény, or error function, is a special function that appears in probability, statistics, and partial differential equations. It is denoted by erf(x). Mathematically, it is defined as the integral of the Gaussian function from negative infinity to x, scaled by a constant. The specific formula involves e^(-t^2), the exponential function, and the integral is taken with respect to a dummy variable t.

The error function is an odd function, meaning that erf(-x) = -erf(x). Its value at x=0 is 0, and as x approaches infinity, the value of erf(x) approaches 1. The error function is closely related to the cumulative distribution function (CDF) of the normal distribution. Specifically, the CDF of a standard normal distribution can be expressed in terms of the error function.

It is particularly useful in solving the heat equation and other diffusion-related problems. The solution to the one-dimensional heat equation with certain boundary conditions often involves the error function. In statistics, it is used to calculate probabilities associated with normally distributed random variables, especially when dealing with ranges that do not correspond to simple multiples of the standard deviation. Due to its importance, tabulated values and computational approximations for the error function are widely available.