2.12 The Error Function 199
Figure 10 Graph of the error function erf(z)for− 3 <z<3.2.12 The Error Function
InSection11wemadetransformationsofaFourierintegraltoobtaintheso-
lution of the heat problem
∂^2 u
∂x^2 =1
k∂u
∂t, −∞<x<∞,^0 <t, (1)
u(x, 0 )=f(x), −∞<x<∞, (2)in the form of a single integral,
u(x,t)=√^1
4 πkt∫∞
−∞f(x′)e−(x−x′)^2 /^4 ktdx′. (3)Even for the simplest functionsf, this integration cannot be carried out in
closed form, mainly because the indefinite integral
∫
e−x^2 dxis not an elemen-
tary function. We can improve our understanding of the solution Eq. (3) if we
introduce theerror function,definedas
erf(z)=√^2
π∫z0e−y^2 dy. (4)Agraphoferf(z)is shown in Fig. 10. Convenient tables, together with approx-
imations to the error function, will be found inHandbook of Mathematical
Functions, by Abramowitz and Stegun.
Several important properties of the error function follow immediately from
the definition. First, it is clear that erf( 0 )=0, and it is easy to show that erf is
an odd function (Exercise 1). Second, by the fundamental theorem of calculus,
the derivative of the error function is
d
dzerf(z)=2
√πe−z^2. (5)