Corollary 5.9.1 . [0544] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Corollary 5.9.1 .
There exists C n > 0 C_{n}>0 z ≥ 1 z\geq 1
(5.136) C n − 1 ⋅ Q − 1 4 − 1 2 n Γ ( Q + 1 ) ⋅ e − j k ⋅ z n 2 + F ( t 0 ( z ) ) ⋅ ( j k z n ) − 1 \displaystyle C_{n}^{-1}\cdot\frac{Q^{-\frac{1}{4}-\frac{1}{2n}}}{\Gamma(Q+1)}\cdot e^{-\frac{j_{k}\cdot z^{n}}{2}+F(t_{0}(z))}\cdot(j_{k}z^{n})^{-1} ≤ 𝒟 k ( z ) ≤ C n ⋅ Q 1 4 Γ ( Q + 1 ) ⋅ e − j k ⋅ z n 2 + F ( t 0 ( z ) ) , \displaystyle\leq\mathcal{D}_{k}(z)\leq C_{n}\cdot\frac{Q^{\frac{1}{4}}}{\Gamma(Q+1)}\cdot e^{-\frac{j_{k}\cdot z^{n}}{2}+F(t_{0}(z))},
(5.137) C n − 1 ⋅ Q − 1 4 ⋅ ( j k ⋅ z n ) 1 − 2 α 4 Γ ( Q + 1 ) ⋅ e − j k ⋅ z n 2 + G ( u 0 ( z ) ) \displaystyle C_{n}^{-1}\cdot Q^{-\frac{1}{4}}\cdot\frac{(j_{k}\cdot z^{n})^{\frac{1-2\alpha}{4}}}{\Gamma(Q+1)}\cdot e^{-\frac{j_{k}\cdot z^{n}}{2}+G(u_{0}(z))} ≤ 𝒢 k ( z ) ≤ C n ⋅ ( j k ⋅ z n ) 1 − 2 α 4 Γ ( Q + 1 ) ⋅ e − j k ⋅ z n 2 + G ( u 0 ( z ) ) , \displaystyle\leq\mathcal{G}_{k}(z)\leq C_{n}\cdot\frac{(j_{k}\cdot z^{n})^{\frac{1-2\alpha}{4}}}{\Gamma(Q+1)}\cdot e^{-\frac{j_{k}\cdot z^{n}}{2}+G(u_{0}(z))},
where Q ≡ α − β − 1 ≥ 1 Q\equiv\alpha-\beta-1\geq 1