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CSS Pure Mathematics QUESTION #1270
Question 1
For \(f(Z)=\dfrac{1}{Z}\), writing \(Z=re^{i\theta}\), the function satisfies the polar C-R equations because:
  • It is not analytic anywhere
  • It is analytic everywhere except \(Z=0\)✔️
  • It is analytic everywhere
  • It is analytic only on the real axis
Correct Answer Explanation
\(f(Z)=\dfrac{1}{Z}=\dfrac{1}{r}e^{-i\theta}=\dfrac{1}{r}(\cos\theta-i\sin\theta)\). So \(u=\dfrac{\cos\theta}{r}\), \(v=-\dfrac{\sin\theta}{r}\). The polar C-R equations are satisfied for all \(r\neq0\), confirming \(f\) is analytic everywhere except at the singularity \(Z=0\).