An Introduction to Thermal Physics · Problem B.6

Systems of Interacting Particles — Daniel Schroeder

Physics — Statistical Mechanics · error-function

Problem

The antiderivative of

e^{-x^2}

, set equal to zero at

x = 0

and multiplied by

2/\sqrt{\pi}

, is called the error function, abbreviated

\operatorname{erf} x

\operatorname{erf} x \equiv \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}\, dt.

(a) Show that

\operatorname{erf}(\pm\infty) = \pm 1

. (b) Evaluate

\displaystyle\int_0^x t^2 e^{-t^2}\, dt

in terms of

\operatorname{erf} x

. (c) Use the result of Problem B.4 to find an approximate expression for

\operatorname{erf} x

when

x \gg 1