The problem involves expressing the derivative of an analytic function $f(z)$ in polar coordinates.

Solution:

The function $z$ in polar form is written as: $z = r e^{i\theta}, \quad \text{where } r = |z| \text{ and } \theta = \arg(z).$

In polar coordinates, the derivative of $f(z)$ (if $f(z)$ is analytic) can be expressed using the chain rule, and the Cauchy-Riemann equations in polar form play a key role. They are: $\frac{\partial f}{\partial r} = \cos\theta \frac{\partial u}{\partial r} - i\sin\theta \frac{\partial u}{\partial r},$ where $u$ and $v$ are the real and imaginary parts of $f(z)$, respectively.

Correct Answer:

From the given options, the correct derivative in polar coordinates is: $\text{(a)} \, (\cos\theta + i\sin\theta)\frac{\partial f}{\partial r}.$

This matches the behavior of $f'(z)$ in the polar representation.

Would you like further explanation about the derivation process? Here are some related questions you might find useful:

1. How are polar coordinates derived for complex numbers?
2. What are the Cauchy-Riemann equations in polar form?
3. Can you explain the significance of $f(z)$ being analytic?
4. How does the derivative $f'(z)$ behave in polar and Cartesian forms?
5. Why is the $e^{i\theta}$ representation of $z$ important in complex analysis?

Tip: When solving problems in polar form, remember to incorporate the chain rule for derivatives involving $r$ and $\theta$.