\[H_{n}\left(x\right)=\frac{(-2i)^{n}{\mathrm{e}}^{x^{2}}}{{\pi}^{\frac{1}{2}}}%
\int_{-\infty}^{\infty}{\mathrm{e}}^{-t^{2}}t^{n}{\mathrm{e}}^{2ixt}\,\mathrm{%
d}t=\frac{2^{n+1}}{{\pi}^{\frac{1}{2}}}{\mathrm{e}}^{x^{2}}\int_{0}^{\infty}{%
\mathrm{e}}^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi\right)\,\mathrm{d}t.\]