\section*{Problem 3}
In the following, $z$ is a complex number and $i$ is the imaginary unit. Answer the following questions on the complex function

$$f ( z ) = \frac { \cot z } { z ^ { 2 } }$$

Here, $\cot z = \frac { 1 } { \tan z }$. For a positive integer $m$, we define

$$D _ { m } = \lim _ { z \rightarrow 0 } \frac { \mathrm {~d} ^ { m } } { \mathrm {~d} z ^ { m } } ( z \cot z )$$

If necessary, you may use $D _ { 2 } = - \frac { 2 } { 3 }$ and

$$\lim _ { z \rightarrow n \pi } \frac { z - n \pi } { \sin z } = ( - 1 ) ^ { n }$$

for an 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 $R = \pi ( 2 M + 1 )$. As shown in Fig.1, for a parameter $t$ that ranges in the interval $- \frac { R } { 2 } \leq t \leq \frac { R } { 2 }$, consider the following four line segments $C _ { k } ( k = 1,2,3,4 )$:

$$\begin{aligned}
& C _ { 1 } : z ( t ) = \frac { R } { 2 } + i t \\
& C _ { 2 } : z ( t ) = - t + i \frac { R } { 2 } \\
& C _ { 3 } : z ( t ) = - \frac { R } { 2 } - i t \\
& C _ { 4 } : z ( t ) = t - i \frac { R } { 2 }
\end{aligned}$$

The initial point of each line segment corresponds to $t = - \frac { R } { 2 }$ and the terminal point corresponds to $t = \frac { R } { 2 }$. For each complex integral $I _ { k } = \int _ { C _ { k } } f ( z ) \mathrm { d } z$ along $C _ { k } ( k = 1,2,3,4 )$, find $\lim _ { M \rightarrow \infty } I _ { k }$.

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 $I = \oint _ { C } f ( z ) \mathrm { d } z$ along the closed loop $C$, find the value of the following infinite series:

$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$$

V. $f ( z )$ is now replaced with the complex function

$$g ( z ) = \frac { \cot z } { z ^ { 2 N } }$$

where $N$ is a positive integer. By following the steps in I-IV, express the following infinite series in terms of $D _ { m }$:

$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 N } }$$