Part CCSIR NET June 2025the-residue-at-an-essential-singularity-convolves-both-series

The residue at an essential singularity convolves both series

Let $γ : [0, 1] \to C$ be the function t ↦ $e^{2 π i t}$ and $I = \int_{γ} e^{z}$ $e^{1/ z}$ dz. Which of the following statements are true?

A.I = 0
B. $(1/2 π i) I \in$ { $4 n : n \in Z, n \geq 1$ }
C. $I = 2 π i \sum_{n = 0}^{\infty}$ 1/n!
D. $I = 2 π i \sum_{n = 0}^{\infty}$ 1/(n!(n + 1)!)✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: ∮γ f = 0 implies f is holomorphic inside γ

The residue at an essential singularity convolves both series

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