Lemma 4.7 . [03HD]

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Lemma 4.7.

For fixed $j\in\mathbb{Z}_{+}$ and $h>0$ , we have the following asymptotic formula

(4.64)

\lim\limits_{z\to+\infty}\frac{\mathcal{F}(z)}{\sqrt{\pi}e^{\frac{jz^{2}}{2}}(\sqrt{j}z)^{h}}=1

and

(4.65)

\lim\limits_{z\to+\infty}\frac{\mathcal{U}(z)}{e^{-\frac{jz^{2}}{2}}\Gamma(h+1)(2\sqrt{j}z)^{-h-1}}=1.