fix: proofread Complex Analysis part

[vEnhance]

Proofreading pass over the Complex Analysis part (issue #315), covering tex/complex-ana/{holomorphic,meromorphic,log,quintic}.tex. Findings are mostly minor: typos, missing factors, and a couple of inverted statements.

holomorphic.tex

• Cauchy's differentiation formula: the coefficient was written $c_k = \frac{f^{k}(p)}{k!}$; corrected the derivative notation to $f^{(k)}(p)$ (consistent with the lines immediately above and below).
• Two references to the differentiation-formula integrand wrote the numerator as $f(w-p)$; corrected to $f(w)$ to match the theorem statement $\frac{1}{2\pi i}\oint_\gamma \frac{f(w)}{(w-p)^{k+1}},dw$.

meromorphic.tex

• "Function with infinitely many poles" example: the function is $\frac{1}{\sin(z)}$ (poles at $z=\pi k$), but the Laurent-series computation was headed $\frac{1}{\sin(2\pi z)}$ while its expansion $\frac{1}{z/1! - z^3/3! + \cdots}$ is that of $\frac{1}{\sin z}$. Removed the spurious $2\pi$.
• Minor grammar fixes in the geometric Argument Principle digression ("shuffle", "mean", "move/follow").

log.tex

• Winding-number integral: restored the missing $i$, $\frac{1}{2\pi}\oint_\gamma \frac{f'}{f},dz \in 2\ZZ \to \frac{1}{2\pi i}\oint_\gamma \frac{f'}{f},dz \in 2\ZZ$ (matches the winding-number definition and Theorem on $n$th roots).
• Worked square-root example: the surrounding text and the rendered diagram use the branch $-\tfrac\pi2 < \alpha \le \tfrac\pi2$, under which $-4 \mapsto 2i$. The example had written the interval as $-\tfrac\pi2 \le \alpha < \tfrac\pi2$ and the image as $-2i$ ("left edge"); made it consistent ($2i$, "right edge").

quintic.tex

• Cubic-formula term: fixed two typos in the discriminant — inner $27a^{d} \to 27a^{2}d$ (matching the outer term) and $(b^{3}-3ac)^3 \to (b^{2}-3ac)^3$ (standard $\Delta_0 = b^2 - 3ac$).
• Definition of solvable: was stated inverted ("if its derived series is nontrivial"); corrected to "if its derived series eventually becomes trivial", which is consistent with the goal of showing $S_5$ is not solvable (its derived series never reaches the trivial group).
• Derived series description: "each group is the commutator subgroup of the next" → "of the previous" ($G^{(k+1)} = [G^{(k)}, G^{(k)}]$).
• Minor grammar fixes ("what happens when we", "paths that don't go", stray comma).

No large mathematical errors were found in this part.

🤖 Generated with Claude Code

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Generated by Claude Code