For a complex function $f(z) = u(r, \theta) + i v(r, \theta)$ to be analytic in polar coordinates $(r, \theta)$, the Cauchy-Riemann equations that $u$ and $v$ must satisfy are:

(A) $\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} \quad \text{and} \quad \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}$

(B) $\frac{\partial u}{\partial r} = r \frac{\partial v}{\partial \theta} \quad \text{and} \quad \frac{\partial v}{\partial r} = -r \frac{\partial u}{\partial \theta}$

(C) $\frac{\partial u}{\partial r} = \frac{\partial v}{\partial \theta} \quad \text{and} \quad \frac{\partial v}{\partial r} = -\frac{\partial u}{\partial \theta}$

(D) $\frac{\partial u}{\partial \theta} = \frac{1}{r} \frac{\partial v}{\partial r} \quad \text{and} \quad \frac{\partial v}{\partial \theta} = -\frac{1}{r} \frac{\partial u}{\partial r}$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles Transforming the Cartesian Cauchy-Riemann equations ($\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$) via chain rule with $x = r \cos\theta$, $y = r \sin\theta$ yields the polar form: $$u_r = \frac{1}{r} v_\theta$$ $$v_r = -\frac{1}{r} u_\theta$$ ### 2. Analytical Verification Differentiating $f(z)$ along radial and transverse increments: $$f'(z) = e^{-i\theta} \left(\frac{\partial u}{\partial r} + i \frac{\partial v}{\partial r}\right) = \frac{e^{-i\theta}}{r} \left(\frac{\partial v}{\partial \theta} - i \frac{\partial u}{\partial \theta}\right)$$ Equating real and imaginary components yields the exact polar relations. ### 3. Distractor Forensics - **Option A:** Correct polar Cauchy-Riemann equations. - **Option B:** Inverted factor ($r$ instead of $1/r$). ### 4. Exam Trap & Standard Reference - **Trap Alert:** Remember the factor $1/r$ on the angle derivative terms. - **Standard Reference:** Churchill, R.V. & Brown, J.W., *Complex Variables and Applications*, Chapter 2.

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