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Part CCSIR NET December 2023residue-of-exp-over-power

Residue of exp over power

For an integer k, consider the contour integral $I_{k} = \int_{∣ z ∣ = 1}$ eᶻ/zᵏ dz. Which of the following statements are true?

A. $I_{k} = 0$ for every integer k.
B. $I_{k} \neq = 0$ if k ≥ 1.✓
C.| $I_{k}$ | ≤ | $I_{k}$ ₊ $_{1}$ | for every integer k.
D. $l i m_{k \to \infty}$ | $I_{k}$ | $= \infty$ .

Solution

$I_{k} = 2 π i \cdot ($ coefficient of $z^{k - 1}$ in eᶻ $) = 2 π i / (k - 1)$ ! for k ≥ 1 and 0 for k ≤ 0. These values decrease to 0, so (3) and (4) fail.

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

From Singularities and Residues › Residue theorem and standard contour integrals

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