Math Problem Statement

Cauchy Riemann equation in polar form

Solution

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

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

x=rcosθ,y=rsinθx = r \cos \theta, \quad y = r \sin \theta

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

ur=1rvθ\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} vr=1ruθ\frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}

These equations ensure that the function f(z)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.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Polar Coordinates
Holomorphic Functions

Formulas

∂u/∂r = (1/r) ∂v/∂θ
∂v/∂r = -(1/r) ∂u/∂θ

Theorems

Cauchy-Riemann Equations

Suitable Grade Level

Undergraduate Mathematics (Advanced Calculus or Complex Analysis)