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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]→C\gamma : [0, 1] \to \mathbb{C} be the function t ↦ e2πite^{2\pi{}it} and I=∫γezI = \int_\gamma e^z e1/ze^{1/z} dz. Which of the following statements are true?

  1. A.I = 0
  2. B.(1/2πi)I∈(1/2\pi{}i)I \in {4n:n∈Z,n≥14n : n \in \mathbb{Z}, n \ge 1}
  3. C.I=2πi∑n=0∞I = 2\pi{}i \sum_{n=0}^\infty 1/n!
  4. D.I=2πi∑n=0∞I = 2\pi{}i \sum_{n=0}^\infty 1/(n!(n + 1)!)✓ Correct

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

More on this topic

The chapter behind this: Residue theorem — the five standard patterns — free to read

From Singularities and Residues › Residue theorem and standard contour integrals

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