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The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}^2}{r^2 + \beta_{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 1} & &[N \geq r] \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x \geq 1 + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that when $n \approx r$, the ratio is approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi$ (asymptotically decreasing to $\pi$) every time $n$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pi)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, which is maximized for $\alpha \in [1,2]$ by $\alpha = 1$, we find that $\frac{\pi^2}{\pi^2 - 1}$ is the bound we want.

The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}^2}{r^2 + \beta_{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 4} \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x \geq 1 + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that as $n$ increases to $r$, the ratio increases to approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi/2$ (asymptotically decreasing to $\pi/2$) every time $n$ increases by $1$ or by a bit more than $\pi$ (asymptotically decreasing ot $\pi$) every time $k$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pi)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, which is maximized for $\alpha \in [1/2,\infty)$ by $\alpha = 1/2$, we find that $\frac{\pi^2}{\pi^2 - 4}$ is a bound we can use. Then summing the geometric series, we have \begin{align} S_K(N,K) &\leq \frac{8 \pi^2}{\pi^2 - 4} \frac{|x| + |r|}{x^4} \frac{\mathrm{e}^{-(N+2K-3)\delta \pi/2} \left( \mathrm{e}^{N \delta \pi / 2} - 1 \right)}{ \left( \mathrm{e}^{\delta \pi / 2} - 1 \right)^2 \left( \mathrm{e}^{\delta \pi/2} + 1 \right)} & &[K \geq 1 + \frac{\sqrt{2}}{\pi} x] \\ S_N(N) &\leq \frac{8 \pi^2}{\pi^2 - 4} \frac{|x| + |r|}{x^4} \frac{\mathrm{e}^{-(N-1) \delta \pi / 2}}{\left( \mathrm{e}^{\delta \pi / 2} - 1 \right)^2 \left( \mathrm{e}^{\delta \pi/2} + 1 \right)} & &[K \geq 1 + \frac{\sqrt{2}}{\pi} x] \text{.} \end{align}

The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}^2}{r^2 + \beta_{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 1} & &[N \geq r] \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that when $n \approx r$, the ratio is approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi$ (asymptotically decreasing to $\pi$) every time $n$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pi)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, which is maximized for $\alpha \in [1,2]$ by $\alpha = 1$, we find that $\frac{\pi^2}{\pi^2 - 1}$ is the bound we want.

The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}^2}{r^2 + \beta_{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 1} & &[N \geq r] \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x \geq 1 + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that when $n \approx r$, the ratio is approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi$ (asymptotically decreasing to $\pi$) every time $n$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pi)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, which is maximized for $\alpha \in [1,2]$ by $\alpha = 1$, we find that $\frac{\pi^2}{\pi^2 - 1}$ is the bound we want.

The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}}{r^2 + \beta+{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 1} & &[N \geq r] \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that when $n \approx r$, the ratio is approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi$ (asymptotically decreasing to $\pi) every time $n$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pir)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, we find that $\frac{\pi^2}{\pi^2 - 1}$ is the bound we want.

The first two we analyze as for the first sum: \begin{align} \mathrm{e}^{-\beta_{nk} \delta} &\leq \mathrm{e}^{(2k+n-3)\pi \delta/2} \text{,} \\ \frac{\beta_{nk}^2}{r^2 + \beta_{nk}^2 - n^2} &\leq \frac{\pi^2}{\pi^2 - 1} & &[N \geq r] \text{, and} \\ \frac{(x J_n'(x) + r J_n(x))^2}{(x^2 - \beta_{nk}^2)^2} &\leq \frac{|x| + |r|}{x^4} & &[K \geq \frac{3-n}{2} + \frac{\sqrt{2}}{\pi} x ] \text{.} \end{align} As before, we used the lower bound on the $\beta$s for the exponential and the exact same estimating for the Bessel functions fraction. The fraction in $\beta$s is a little trickier. Note that when $n \approx r$, the ratio is approximately $1$. Subsequently, the $\beta$s increase by a bit more than $\pi$ (asymptotically decreasing to $\pi$) every time $n$ increases by $1$. Looking at $\lim_{k \rightarrow \infty} \frac{(\beta + k \alpha \pi)^2}{r^2 + (\beta + k \alpha \pi)^2 - (r+k)^2} = \frac{\pi^2}{\pi^2 - (1/\alpha)^2}$, which is maximized for $\alpha \in [1,2]$ by $\alpha = 1$, we find that $\frac{\pi^2}{\pi^2 - 1}$ is the bound we want.