Math Problem Statement
Cauchy Riemann equation in polar form
Solution
The Cauchy-Riemann equations in polar form describe the conditions for a complex function to be holomorphic when expressed in polar coordinates.
Given a complex function , where and are real-valued functions representing the real and imaginary parts of , we convert to polar coordinates using the relations:
So, . The Cauchy-Riemann equations in polar form are:
These equations ensure that the function is differentiable and thus holomorphic in polar coordinates.
Would you like more details or any clarifications?
Here are some questions to expand the topic:
- How are the Cartesian form of Cauchy-Riemann equations derived?
- What is the interpretation of holomorphic functions in polar coordinates?
- How do you convert between Cartesian and polar forms of complex functions?
- Can you provide examples of functions that satisfy the Cauchy-Riemann equations in polar form?
- 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)
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