Appendix B

Definite Integrals – Maxwellian Distribution Functions

Evaluation and values of definite integrals related to Maxwellian distribution functions

Source: Vincenti and Kruger (1965), Introduction to Physical Gas Dynamics, New York: Wiley.

A definite integral that appears frequently in kinetic theory and statistical mechanics is the following:

$I_{n} (a) \equiv \int_{0}^{\infty} x^{n} e^{- a x^{2}} d x,$

where a > 0 and n is a nonnegative integer. Values of this integral for specific values of n are

$\begin{matrix} I_{0} (a) = \frac{1}{2} {(\frac{π}{a})}^{\frac{1}{2}}, I_{1} (a) = \frac{1}{2 a}, \\ I_{2} (a) = \frac{1}{4} {(\frac{π}{a^{3}})}^{\frac{1}{2}}, I_{3} (a) = \frac{1}{2 a^{2}}, \\ I_{4} (a) = \frac{3}{8} {(\frac{π}{a^{5}})}^{\frac{1}{2}}, I_{5} (a) = \frac{1}{a^{3}}, etc . \end{matrix}$

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