\[{\operatorname{ber}}^{2}x+{\operatorname{bei}}^{2}x\sim\frac{e^{x\sqrt{2}}}{2%
\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^{2}}-\frac%
{33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{8192}\frac{1}{x^{4}}+\dotsb\right),\]