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GATE 2023 PH – Question 56

Mathematical Physics · complex analysis: Cauchy-Riemann conditions, Cauchy's theorem, singularities, residue theorem and applications · 2 marks · Multiple select

For $f(z)=z^2\sin z/(z-\pi)^4$, which statements at $z=\pi$ are correct?

  1. The pole order is 4
  2. The pole order is 3
  3. The residue is $\pi/6$
  4. The residue is $2\pi/3$

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Correct answer: (B) The pole order is 3

Explanation

Let $w=z-\pi$, so $z=\pi+w$ and $\sin z=\sin(\pi+w)=-\sin w=-w+\dfrac{w^3}{6}-\cdots$.

$$f(z)=\frac{z^2\sin z}{(z-\pi)^4}=\frac{(\pi+w)^2\left(-w+\frac{w^3}6-\cdots\right)}{w^4}=-\frac{(\pi+w)^2}{w^3}+\frac{(\pi+w)^2}{6w}-\cdots$$

**Order of the pole.** The most negative power of $w$ is $w^{-3}$ (with coefficient $-\pi^2\neq0$), so the pole is of **order 3**. Statement B is true; statement A is false.

**Residue** (coefficient of $w^{-1}$):
- from $-\dfrac{(\pi+w)^2}{w^3}=-\dfrac{\pi^2+2\pi w+w^2}{w^3}$: the $w^{-1}$ term is $-1$;
- from $\dfrac{(\pi+w)^2}{6w}$: the $w^{-1}$ term is $\dfrac{\pi^2}{6}$.

$$\text{Res}=\frac{\pi^2}{6}-1 .$$

This is neither $\pi/6$ nor $2\pi/3$, so statements C and D are false.

Answer **B** only.