Question

Find the Cauchy-Riemann equations in polar coordinates. Hint : \(z=r e^{i \theta}, f(z)=u(r, \theta)+i \tau(r, \theta) .\) Follow the method of equations \((2.3)\) and \((2.4) .\)

Short answer

The Cauchy-Riemann equations in polar coordinates are \( \frac{\text{∂u}}{\text{∂r}} = \frac{1}{r}\frac{\text{∂τ}}{\text{∂θ}} \) and \( \frac{\text{∂τ}}{\text{∂r}} = -r\frac{\text{∂u}}{\text{∂θ}}. \)

Step-by-step solution

Express the Function in Polar Coordinates

Given the function in Cartesian coordinates as f(z) = u(x, y) + i v(x, y), convert it using the polar coordinates where z = r e^{iθ}. This implies using f(z) = u(r, θ) + i τ(r, θ).

Define Partial Derivatives

Express the partial derivatives of u and v in terms of r and θ: \( \frac{\text{∂u}}{\text{∂r}} \text{,} \frac{\text{∂u}}{\text{∂θ}} \text{,} \frac{\text{∂τ}}{\text{∂r}} \text{, and} \frac{\text{∂τ}}{\text{∂θ}}.\)

Convert Cartesian Cauchy-Riemann Equations

Convert the equations from Cartesian to polar coordinates. The Cartesian Cauchy-Riemann equations are: \( \frac{\text{∂u}}{\text{∂x}} = \frac{\text{∂v}}{\text{∂y}}, \) and \( \frac{\text{∂u}}{\text{∂y}} = -\frac{\text{∂v}}{\text{∂x}}. \)

Use Chain Rule for Partial Derivatives

Express the partial derivatives of x and y with respect to r and θ using the chain rule: \( x = r \text{cosθ}, \) and \( y = r \text{sinθ}. \) So, \( \frac{\text{∂u}}{\text{∂x}} = \frac{\text{∂u}}{\text{∂r}} \frac{\text{∂r}}{\text{∂x}} + \frac{\text{∂u}}{\text{∂θ}} \frac{\text{∂θ}}{\text{∂x}}, \) and similarly for other partial derivatives.

Simplify Using Polar Coordinates

Substitute the partial derivatives of r and θ with respect to x and y into the equations: \( \frac{\text{∂r}}{\text{∂x}} = \text{cosθ}, \) \( \frac{\text{∂r}}{\text{∂y}} = \text{sinθ}, \) \( \frac{\text{∂θ}}{\text{∂x}} = -\frac{\text{sinθ}}{r}, \) and \( \frac{\text{∂θ}}{\text{∂y}} = \frac{\text{cosθ}}{r}. \)

Derive the Cauchy-Riemann Equations in Polar Form

Combine and simplify the equations to get: \[ \frac{\text{∂u}}{\text{∂r}} = \frac{1}{r}\frac{\text{∂τ}}{\text{∂θ}}, \] \[ \frac{\text{∂τ}}{\text{∂r}} = -r\frac{\text{∂u}}{\text{∂θ}}. \]

Key concepts

complex functions

Complex functions are functions that take complex numbers as inputs and provide complex numbers as outputs. A complex number is expressed in the form: \( z = x + iy \), where \( x \) is the real part and \( iy \) is the imaginary part.
In the context of the Cauchy-Riemann equations, we often express these functions as \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) \) and \( v(x, y) \) represent the real and imaginary parts of the function, respectively.
When considering complex functions in polar coordinates, we utilize the transformation \( z = re^{i\theta} \), which converts the function \( f(z) \) to a form involving \( r \) and \( \theta \). Typically, this transformation leads to an expression like \( f(z) = u(r, \theta) + i\tau(r, \theta) \), enabling us to explore the behavior of complex functions more easily in different coordinate systems.

partial derivatives

Partial derivatives are essential when studying the behavior of multivariable functions, like \( f(x, y) \). They measure how a function changes as the input varies along one of its dimensions while holding other dimensions constant.
For complex functions, we often deal with the derivatives \( \frac{\text{∂u}}{\text{∂x}} \) and \( \frac{\text{∂v}}{\text{∂y}} \) in Cartesian coordinates. But when we switch to polar coordinates, we look at \( \frac{\text{∂u}}{\text{∂r}} \), \( \frac{\text{∂u}}{\text{∂\theta}} \), \( \frac{\text{∂\tau}}{\text{∂r}} \), and \( \frac{\text{∂\tau}}{\text{∂\theta}} \).
The chain rule helps us convert the expressions for partial derivatives between different coordinate systems. For instance, knowing that \( x = r\text{cos}\theta \) and \( y = r\text{sin}\theta \), we can express the Cartesian partial derivatives in terms of polar coordinates.

polar coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance \( r \) from a reference point and an angle \( \theta \) from a reference direction.
Polar coordinates provide a useful alternative to Cartesian coordinates, especially in problems exhibiting radial symmetry. The transformation from Cartesian to polar coordinates is given by:
\( x = r \text{cos} \theta \)
\( y = r \text{sin} \theta \)
Using these relations, we can express complex functions and their derivatives in polar form. This transformation simplifies analyzing problems involving circular or spiral patterns.
In deriving the Cauchy-Riemann equations in polar coordinates, we convert the Cartesian form of the equations, making use of substitutions and simplifications to obtain:
\( \frac{\text{∂u}}{\text{∂r}} = \frac{1}{r} \frac{\text{∂\tau}}{\text{∂\theta}} \)
\( \frac{\text{∂\tau}}{\text{∂r}} = -r \frac{\text{∂u}}{\text{∂\theta}} \).