One definition of the Bessel-K function is the integral

\[
\int_{0}^{\infty} \frac{\cos (\omega t) d \omega}{\left(1+\omega^{2}ight)^{v+1 / 2}}=\frac{\sqrt{ } \pi}{2^{v} \Gamma(v+1 / 2)} \times|t|^{v} \mathcal{K}_{v}(t)
\]

Deduce that \(\mathcal{K}_{v}(\cdot)\) is symmetric and that

\[
\lim _{t ightarrow 0}|t|^{v} \mathcal{K}_{v}(|t|)=2^{u-1} \Gamma(u) .
\]