The Basic Gaussian Integral
The function^2 "(e^-^x)^2")
, called a Gaussian, appears everywhere in the mathematical sciences. It plays a fundamental role in probability and statistics.
The fundamental Gaussian integral in its simplest form
If follow when a is positive constant. In fact this remains true even if a is a complex number, as long as Re a > 0
For x ∈ R, the error function is defined by
&space;=&space;\frac{2}{\pi}\int_{0}^{x&space;}&space;e^-^{t^2}&space;dt "erf (x) = \frac{2}{\pi}\int_{0}^{x } e^-^{t^2} dt")
The gaussian function, error function and complementary error function are frequently used in probability theory since the normalized gaussian curve represents the probability distribution with standard deviation s relative to the average of a random distribution. The error function represents the probability that the parameter of interest is within a range between -x/sÖ2and +x/sÖ2 while the complementary error function provides the probability that the parameter is outside that range
Relations and Selected Values of Error Functions
erf (∞) = 1
erf (−x) = − erf (x)
erf (0) = 0
erf (−∞) = −1
The function
If follow when a is positive constant. In fact this remains true even if a is a complex number, as long as Re a > 0
For x ∈ R, the error function is defined by
The gaussian function, error function and complementary error function are frequently used in probability theory since the normalized gaussian curve represents the probability distribution with standard deviation s relative to the average of a random distribution. The error function represents the probability that the parameter of interest is within a range between -x/sÖ2and +x/sÖ2 while the complementary error function provides the probability that the parameter is outside that range
Relations and Selected Values of Error Functions
erf (∞) = 1
erf (−x) = − erf (x)
erf (0) = 0
erf (−∞) = −1
