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Lemma 5.8 .
Let y ≤ − 1 y\leq-1
(5.98) C n − 1 ⋅ e y ( − y ) 1 − 2 α 4 Γ ( α − β ) ⋅ ∫ 1 − y ∞ e G ( u ) 𝑑 u ≤ Φ ♯ ( β , α , y ) ≤ C n ⋅ e y ( − y ) 1 − 2 α 4 Γ ( α − β ) ⋅ ∫ 0 ∞ e G ( u ) du , C_{n}^{-1}\cdot\frac{e^{y}(-y)^{\frac{1-2\alpha}{4}}}{\Gamma(\alpha-\beta)}\cdot\int_{\frac{1}{\sqrt{-y}}}^{\infty}e^{G(u)}du\leq\Ku(\fb,\fa,y)\leq C_{n}\cdot\frac{e^{y}(-y)^{\frac{1-2\alpha}{4}}}{\Gamma(\alpha-\beta)}\cdot\int_{0}^{\infty}e^{G(u)}du,
where
(5.99) G ( u ) ≡ − u 2 + 2 − y u + ( α − 2 β − 1 2 ) log u . G(u)\equiv-u^{2}+2\sqrt{-y}u+(\alpha-2\beta-\frac{1}{2})\log u.