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Lemma 5.7 .
For each fixed k k z → + ∞ z\to+\infty
(5.93) 𝒢 k ( z ) \displaystyle\mathcal{G}_{k}(z) ∼ Γ ( α ) Γ ( α − β ) ⋅ ( j k z n ) − β ⋅ e j k z n 2 , \displaystyle\sim\frac{\Gamma(\alpha)}{\Gamma(\alpha-\beta)}\cdot(j_{k}z^{n})^{-\beta}\cdot e^{\frac{j_{k}z^{n}}{2}},
(5.94) 𝒟 k ( z ) \displaystyle\mathcal{D}_{k}(z) ∼ ( j k z n ) β − α ⋅ e − j k z n 2 . \displaystyle\sim(j_{k}z^{n})^{\beta-\alpha}\cdot e^{-\frac{j_{k}z^{n}}{2}}.