Problem Analysis #2

538.47s

74,144 toks

Problem Statement

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 $\mathcal{K}= 999\,(p^2 - 1)\,(-3r)\,(-D_2)\,\frac{S_2}{T}.$ \

Ground Truth

5328

Execution Trace

Reasoning Process

We first determine the required quantities from the function (f(z)=\frac{\cot z}{z^2}).

Pole at (z=0):
(\cot z = \frac{1}{z} - \frac{z}{3} - \frac{z^3}{45} - \cdots)
(\Rightarrow f(z) = \frac{1}{z^3} - \frac{1}{3z} + \cdots)
Hence the order of the pole at (z=0) is (p = 3), and the residue is (r = -\frac{1}{3}).

Given: (D_2 = -\dfrac{2}{3}).

Sum (\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2} = S_2):
From the residue theorem applied to the square contour in part IV, one obtains (S_2 = \frac{\pi^2}{6}).

General sum for (g(z)=\frac{\cot z}{z^{2N}}):
By the same contour method,
[ \sum_{n=1}^{\infty}\frac{1}{n^{2N}} = -\frac{\pi^{2N} D_{2N}}{2,(2N)!}. ]
For (N=1) this gives
[ T = -\frac{\pi^{2} D_2}{2\cdot 2!} = -\frac{\pi^{2} D_2}{4} = -\frac{\pi^{2}}{4}\left(-\frac{2}{3}\right) = \frac{\pi^{2}}{6}. ]
Thus (S_2 = T), so (\displaystyle\frac{S_2}{T}=1).

Compute (\mathcal{K}):
[ \begin{aligned} p^2-1 &= 3^2 - 1 = 8,\ -3r &= -3\left(-\frac{1}{3}\right) = 1,\ -D_2 &= -\left(-\frac{2}{3}\right) = \frac{2}{3},\ \frac{S_2}{T} &= 1. \end{aligned} ]
[ \mathcal{K} = 999 \times 8 \times 1 \times \frac{2}{3} \times 1 = 999 \times \frac{16}{3} = 999 \times \frac{16}{3} = 333 \times 16 = 5328. ]

[ \boxed{5328} ]

Reasoning Process

[THOUGHT]
We compute K using the formula:
K = 999 * (p2 - 1) * (-3 * r) * (-D_2) * (S_2 / T)
with p = 3, r = -1/3, D_2 = -2/3, S_2 = pi2/6, T = pi**2/6.
We'll compute in Python for verification.

[/THOUGHT] [PYTHON] import math

p = 3 r = -1/3 D2 = -2/3 S2 = math.pi2 / 6 T = math.pi2 / 6

K = 999 * (p**2 - 1) * (-3 * r) * (-D2) * (S2 / T) print(K) [/PYTHON]

Tool Output

[PYTHON OUTPUT] 5328.0 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT]
The problem requires computing the integer $\mathcal{K} = 999\,(p^2 - 1)\,(-3r)\,(-D_2)\,\frac{S_2}{T}$ , where:

$p = 3$ (order of the pole of $f(z)$ at $z=0$ ),
$r = -\frac{1}{3}$ (residue of $f(z)$ at $z=0$ ),
$D_2 = -\frac{2}{3}$ (given),
$S_2 = \frac{\pi^2}{6}$ (value of $\sum_{n=1}^\infty \frac{1}{n^2}$ from part IV),
$T = \frac{\pi^2}{6}$ (value of $\sum_{n=1}^\infty \frac{1}{n^2}$ from part V with $N=1$ ).

Since $S_2 = T$ , their ratio is $1$ . Substituting values:

\mathcal{K} = 999 \cdot (3^2 - 1) \cdot \left(-3 \cdot \left(-\frac{1}{3}\right)\right) \cdot \left(-(-\frac{2}{3}\right) \cdot 1

Simplifying step-by-step:

$p^2 - 1 = 8$ ,
$-3r = -3 \cdot (-\frac{1}{3}) = 1$ ,
$-D_2 = -(-\frac{2}{3}) = \frac{2}{3}$ .

Thus:

\mathcal{K} = 999 \cdot 8 \cdot 1 \cdot \frac{_so I'll compute this in Python to confirm. </THOUGHT> [PYTHON] import math p = 3 r = -1/3 D2 = -2/3 S2 = math.pi**2 / 6 T = math.pi**2 / 6 K = 999 * (p**2 - 1) * (-3 * r) * (-D2) * (S2 / T) K [/PYTHON] 5328.0

Tool Output

[PYTHON OUTPUT] 5328.0 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT]
We need to compute the integer
[ \mathcal{K} = 999,(p^2 - 1),(-3r),(-D_2),\frac{S_2}{T} ]
with

(p = 3) (order of pole of (f(z) = \frac{\cot z}{z^2}) at (z=0))
(r = -\frac{1}{3}) (residue of (f(z)) at (z=0))
(D_2 = -\frac{2}{3}) (given)
(S_2 = \frac{\pi^2}{6}) (sum from part IV)
(T = \frac{\pi^2}{6}) (sum from part V with (N=1))

Since (S_2 = T), (\frac{S_2}{T} = 1).

Now compute step-by-step:
[ p^2 - 1 = 9 - 1 = 8 ]
[ -3r = -3 \cdot \left(-\frac{1}{3}\right) = 1 ]
[ -D_2 = -\left(-\frac{2}{3}\right) = \frac{2}{3} ]
Thus
[ \mathcal{K} = 999 \cdot 8 \cdot 1 \cdot \frac{2}{3} \cdot 1 = 999 \cdot \frac{16}{3} = 333 \cdot 16 = 5328 ]

We verify with Python:

</THOUGHT> [PYTHON] import math

p = 3 r = -1/3 D2 = -2/3 S2 = math.pi2 / 6 T = math.pi2 / 6

K = 999 * (p**2 - 1) * (-3 * quest: I realize I made a small error in the Python code block in the previous thought. I'll correct it now.

In the previous step, I had a Python block, but I need to ensure it's correct. Let me write the correct code:

[THOUGHT] We have:

p = 3
r = -1/3
D2 = -2/3
S2 = math.pi**2 / 6
T = math.pi**2 / 6

Compute K = 999 * (p**2 - 1) * (-3 * r) * (-D2) * (S2 / T)

Let's compute this in Python. </think>

[THOUGHT] We compute K using the given values. </THOUGHT> [PYTHON] import math

p = 3 r = -1/3 D2 = -2/3 S2 = math.pi2 / 6 T = math.pi2 / 6

K = 999 * (p**2 - 1) * (-3 * r) * (-D2) * (S2 / T) print(K) [/PYTHON]

Tool Output

[PYTHON OUTPUT] SyntaxError: unterminated string literal (detected at line 9) (<string>, line 9) [/PYTHON OUTPUT]

Reasoning Process

5328

Reasoning Process

[THOUGHT] The integer 𝒦 = 5328.

\boxed{5328}

Reasoning Process

[THOUGHT] The integer (\mathcal{K} = 999 \times (p^2 - 1) \times (-3r) \times (-D_2) \times \frac{S_2}{T}), with (p = 3), (r = -\frac{1}{3}), (D_2 = -\frac{2}{3}), (S_2 = \frac{\pi^2}{6}), (T = \frac{\pi^2}{6}).
Since (S_2 / T = 1), we compute:

(p^2 - 1 = 8)
(-3r = -3 \times (-\frac{1}{3}) = 1)
(-D_2 = -(-\frac{2}{3}) = \frac{2}{3}) Thus (\mathcal{K} = 999 \times 8 \times 1 \times \frac{2}{3} = 999 \times \frac{16}{3} = 333 \times 16 = 5328).

[ \boxed{5328} ]

Reasoning Process

[THOUGHT] We'll compute K using Python. [/THOUGHT] [PYTHON] import math

p = 3 r = -1/3 D2 = -2/3 S2 = math.pi2 / 6 T = math.pi2 / 6

K = 999 * (p**2 - 1) * (-3 * r) * (-D2) * (S2 / T) K [/PYTHON]

Tool Output

[PYTHON OUTPUT] 5328.0 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The integer (\mathcal{K} = 5328). [ \boxed{5328} ]

Final Model Output

5328

Comparative Analysis (P2) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

448.4

Solar-Open-100B

223.1

Llama-VARCO-8B-Instruct

149.6

Gemini-3-Pro-Preview

144.7

Kanana-2-30B-Thinking

97.9

Grok-4.1-fast

89.6

K-EXAONE-236B-A23B

72.4

Solar-Pro-2 (31B)(high)

71.1

GPT-5.1 (high)

55.4

Deepseek-V3.2

48.1

A.X-4.0 (72B)

43.9

Claude-Opus-4.5

33.5

Kanana-2-30B-Thinking-2601

33.5

HCX-007(high)

32.2

axk1

18.0

EXAONE-4.0.1-32B (high)

15.1

GPT-5.2 (high)

11.9

Avg Token Usage

K-EXAONE-236B-A23B

57849.7

Solar-Open-100B

36387.0

Kanana-2-30B-Thinking

18183.0

Grok-4.1-fast

14348.0

Solar-Pro-2 (31B)(high)

13266.0

Llama-VARCO-8B-Instruct

12669.0

K-EXAONE-236B-A23B

11254.0

Gemini-3-Pro-Preview

10743.0

Kanana-2-30B-Thinking-2601

9313.5

Deepseek-V3.2

8345.0

Claude-Opus-4.5

8331.0

EXAONE-4.0.1-32B (high)

8303.0

GPT-5.1 (high)

5813.0

axk1

5525.0

HCX-007(high)

5438.0

GPT-5.2 (high)

4500.0

A.X-4.0 (72B)

3825.0