The **Cauchy-Riemann equations** in polar form describe the conditions for a complex function \( f(z) \) to be holomorphic when expressed in polar coordinates.

Given a complex function \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) \) and \( v(x, y) \) are real-valued functions representing the real and imaginary parts of \( f \), we convert to polar coordinates using the relations:

\[
x = r \cos \theta, \quad y = r \sin \theta
\]

So, \( f(z) = f(r, \theta) = u(r, \theta) + iv(r, \theta) \). The **Cauchy-Riemann equations** in polar form are:

\[
\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}
\]
\[
\frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}
\]

These equations ensure that the function \( f(z) \) is differentiable and thus holomorphic in polar coordinates.

Would you like more details or any clarifications?

Here are some questions to expand the topic:
1. How are the Cartesian form of Cauchy-Riemann equations derived?
2. What is the interpretation of holomorphic functions in polar coordinates?
3. How do you convert between Cartesian and polar forms of complex functions?
4. Can you provide examples of functions that satisfy the Cauchy-Riemann equations in polar form?
5. How do the Cauchy-Riemann equations relate to harmonic functions?

**Tip:** To check if a function is analytic, ensure that both the Cauchy-Riemann equations and differentiability conditions are satisfied.

Understand the Cauchy-Riemann equations in polar form, crucial for determining when a complex function is holomorphic. This page explains the conversion to polar coordinates and provides key formulas for both the real and imaginary parts of the function. Suitable for advanced undergraduate mathematics students.

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