GATE 2024 PH – Question 31

Mathematical Physics · complex analysis: Cauchy-Riemann conditions, Cauchy's theorem, singularities, residue theorem and applications · 1 mark · Multiple select

The complex function $e^{-2/(z-1)}$ has

a simple pole at $z=1$
an essential singularity at $z=1$
residue −2 at $z=1$
a branch point at $z=1$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) an essential singularity at $z=1$; (C) residue −2 at $z=1$

Explanation

Let $w=z-1$. Expand the exponential in a series:
$$e^{-2/(z-1)}=e^{-2/w}=\sum_{n=0}^\infty\frac{1}{n!}\left(-\frac2w\right)^n=1-\frac{2}{w}+\frac{2}{w^2}-\cdots$$

The Laurent series has **infinitely many negative powers** of $(z-1)$.

**A. Simple pole at $z=1$.** False: that needs only one negative power.
**B. Essential singularity at $z=1$.** True: infinitely many negative powers define an essential singularity. ✓
**C. Residue $-2$ at $z=1$.** True: the coefficient of $(z-1)^{-1}$ is $-2$. ✓
**D. Branch point at $z=1$.** False: the function is single valued around $z=1$.

Answer **B and C**.