For f(z) = frac 1 z , writing z = re^ itheta , the Cauchy-Riemann equations in polar form are satisfied when:

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SSC Pure Mathematics QUESTION #4125

Question 1

For $f(z) = \frac{1}{z}$, writing $z = re^{i\theta}$, the Cauchy-Riemann equations in polar form are satisfied when:

$u_r = \frac{1}{r}v_\theta$ and $v_r = -\frac{1}{r}u_\theta$
$u_r = \frac{1}{r}u_\theta$ and $v_r = \frac{1}{r}v_\theta$
$ru_r = v_\theta$ and $rv_r = -u_\theta$
Both (A) and (C) are equivalent correct forms✔️

Correct Answer Explanation

Polar C-R equations: $\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}$ and $\frac{\partial v}{\partial r} = -\frac{1}{r}\frac{\partial u}{\partial \theta}$, equivalent to $ru_r = v_\theta$ and $rv_r = -u_\theta$.

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