Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSwhich is the required result. ◭We note that is it conventional to define, in addition, the functionsγ(a, x)Γ(a, x)P (a, x) ≡ , Q(a, x) ≡Γ(a) Γ(a) ,which are also often called incomplete gamma functions; it is clear that Q(a, x) =1 − P (a, x).18.12.4 The error functionFinally, we mention the error function, which is encountered in probability theoryand in the solutions of some partial differential equations. The error function isrelated to the incomplete gamma function by erf(x) =γ( 1 2 ,x2 )/ √ π and is thusgiven byerf(x) = √ 2 ∫ xe −u2 du =1− 2 ∫ ∞√ e −u2 du. (18.167)π πFrom this definition we can easily see that0erf(0) = 0, erf(∞) =1, erf(−x) =−erf(x).By making the substitution y = √ 2u in (18.167), we find√ ∫ √22xerf(x) = e −y2 /2 dy.π 0The cumulative probability function Φ(x) for the standard Gaussian distribution(discussed in section 30.9.1) may be written in terms of the error function asfollows:Φ(x) = √ 1 ∫ xe −y2 /2 dy2π −∞= 1 2 + √ 1 ∫ xe −y2 /2 dy2π 0= 1 2 + 1 ( ) x√22 erf .It is also sometimes useful to define the complementary error functionerfc(x) =1− erf(x) = √ 2 ∫ ∞e −u2 du = Γ( 1 2 ,x2 )√ . (18.168)π πxx18.1 Use the explicit expressions18.13 Exercises640