\[ \begin{align} \int_{0}^{\infty} a(t)\, dt &= \int_{0}^{\infty} \frac{m}{\left( 2\pi t / \tau \right)^{n/2}} \exp\!\left( -\frac{\tau}{2 t} \right) \, dt \\[10pt] &= \int_{\infty}^{0} \frac{m}{\left( 2\pi (\tau/2u)/\tau \right)^{n/2}} \exp(-u)\, \left( -\frac{\tau}{2u^{2}} \right)\, du \\[10pt] &\left( u = \frac{\tau}{2t} \quad\Rightarrow\quad t = \frac{\tau}{2u},\qquad dt = -\,\frac{\tau}{2u^{2}}\, du \right) \\[10pt] &= \int_{0}^{\infty} \frac{m}{\left( \pi/u \right)^{n/2}} \exp(-u)\, \frac{\tau}{2u^{2}}\, du \\[10pt] &= \frac{m \tau}{2 \pi^{n/2}} \int_{0}^{\infty} u^{n/2-2} \exp(-u)\, du \\[10pt] &= \begin{cases} diverges, & n \leq 2, \\[10pt] \frac{m \tau}{2 \pi^{n/2}} \Gamma(n/2-1), & n \gt 2, \\[10pt] m \tau /{2 \pi}, & n=3. \end{cases} \end{align} \]