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CSS Pure Mathematics QUESTION #4126
Question 1
For $f(z) = \frac{1}{z} = \frac{\cos(\theta)}{r} - i\frac{\sin(\theta)}{r}$ in polar form, compute $u_r$ and verify C-R equations. What is $u_r$?
  • $-\frac{\cos(\theta)}{r^2}$✔️
  • $-\frac{1}{r^2}$
  • $\frac{\cos(\theta)}{r}$
  • $\frac{1}{r}$
Correct Answer Explanation
$u = \frac{\cos(\theta)}{r}$, so $u_r = -\frac{\cos(\theta)}{r^2}$. Also $v = -\frac{\sin(\theta)}{r}$, so $v_\theta = -\frac{\cos(\theta)}{r}$. Then $\frac{1}{r}v_\theta = -\frac{\cos(\theta)}{r^2} = u_r$. C-R satisfied.