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I have been trying to read Jerison and Kenig's article Unique continuation and absence of positive eigenvalues for Schrödinger operators, and I am having difficulties understanding how they obtain the following bound on a product of gamma functions.
The constants are defined throughout the paper, but I'm also writing them here for clarity. Let $t>0$ be such that $m-1<t<m$ for an integer $m$, and let $\delta$ denote the distance from $t$ to the nearest integer. Let $\varepsilon \in (0,\delta/2)$ and $k\in \mathbb{Z}$ be arbitrary. Also arbitrary are $\gamma,\eta\in \mathbb{R}$. Finally, fix a dimension $n\geq3$. On page 469, in the proof of Lemma 2.3, they write the following.
Using Stirling's formula and the functional equation $\Gamma(\zeta)\Gamma(1-\zeta)=\pi/\sin(\pi\zeta)$, it is easy to see that for $0<\varepsilon<\delta/2$ and $\delta$ the distance from $t$ to the nearest integer, $$\left|\frac{\Gamma(\tfrac{1}{2}(k-t-i\eta))\Gamma(\tfrac{1}{2}(n+k+t-\varepsilon-i\gamma+i\eta))}{\Gamma(\tfrac{1}{2}(n-\varepsilon-i\gamma))\Gamma(\tfrac{1}{2}(k-t+\varepsilon+i\gamma-i\eta))\Gamma(\tfrac{1}{2}(n+k+t+i\eta))}\right|\leq Ce^{C|{\gamma}|},$$ where the constant $C$ depends only on $\delta$ and $n$.
No matter what I try, I cannot seem to get such a bound on $\Gamma$. From Stirling's formula I know that: $$\Gamma(z)=\sqrt{2\pi}z^{z-1/2}e^{-z}e^{J(z)},\quad \text{Re }z\geq c >0,$$ where $J$ is a bounded function, uniformly bounded away from the imaginary axis (thus the c>0). Following the authors, it seems that I should set $c=\delta/2$, or something that only depends on $\delta$. Clearly the $J(z)$ terms are not problematic (as long as the Stirling formula applies). Also, all the terms of the form $e^{-z}$ are not problematic, since the arguments of all the gamma functions add up (with sign) to $$e^{\sum_{i=1}^5 \pm z_i}=e^{\tfrac{1}{2}n+\varepsilon +i\gamma},$$ (if my algebra is correct), which gives me the right dependence. What I'm left with are the "ugly" terms of the form $e^{(z-1/2)\text{Log}(z)}$, where the principal branch of the logarithm is chosen. My question is now two-fold:
- Is there a way to tackle the "ugly" terms with log?
- Is it even possible to apply Stirling's formula, when $(k-t)$ in the first Gamma function above might be less than $\delta/2$? (here the obvious answer is that I should apply the reflection formula, but having been at this problem for this long has made me doubt everything...)