Problem Analysis #2

In the following, $z$ is a complex number and $i$ is the imaginary unit. Consider the complex function$f(z) = \frac{\cot z}{z^2},$where $\cot z = \dfrac{1}{\tan z}$. For a positive integer $m$, define$D_m = \lim_{z\to 0} \frac{d^m}{dz^m}(z \cot z).$If necessary, you may use $D_2 = -\dfrac{2}{3}$ and$\lim_{z\to n\pi} \frac{z - n\pi}{\sin z} = (-1)^n\quad\text{for any integer } n.$

• [I.] Find all poles of $f(z)$. Also, find the order of each pole.
• [II.] Find the residue of each pole found in I.
• [III.] Let $M$ be a positive integer and set $R = \pi(2M+1)$. For each real parameter $t$ with $-\dfrac{R}{2} \le t \le \dfrac{R}{2}$, consider the four line segments \begin{align*} C_1:\ & z(t) = \frac{R}{2} + it, \\ C_2:\ & z(t) = -t + i\frac{R}{2}, \\ C_3:\ & z(t) = -\frac{R}{2} - it, \\ C_4:\ & z(t) = t - i\frac{R}{2}. \end{align*} These four oriented segments form the boundary of a square centered at the origin, traversed counterclockwise. For each complex integral $I_k = \displaystyle\int_{C_k} f(z)\,dz$ along $C_k$ ($k = 1,2,3,4$), find $\displaystyle\lim_{M\to\infty} I_k$.
• [IV.] Let $C$ be the closed loop composed of the four line segments $C_1, C_2, C_3$, and $C_4$ in III. By applying the residue theorem to the complex integral $I = \oint_C f(z)\,dz,$ find the value of the infinite series $\sum_{n=1}^{\infty} \frac{1}{n^2}.$
• [V.] Now replace $f(z)$ by the complex function $g(z) = \frac{\cot z}{z^{2N}},$ where $N$ is a positive integer. By following the same contour method as in I–IV, express the infinite series $\sum_{n=1}^{\infty} \frac{1}{n^{2N}}$ in terms of $D_m$. Finally, let $p$ be the order of the pole of $f(z)$ at $z=0$ (from I), and let $r$ be the residue of $f(z)$ at $z=0$ (from II). Let $S_2$ denote the value of $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ obtained in IV. From the expression in V, let $T$ be the value of $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ obtained by setting $N=1$ in your general formula.Compute the integer$\mathcal{K}= 999\,(p^2 - 1)\,(-3r)\,(-D_2)\,\frac{S_2}{T}.$\