Cauchy-Riemann Equations
The Equations
Sufficient Conditions
Wirtinger Derivatives
The Cauchy-Riemann equations can be written as .
Relationship Between and
If is holomorphic, then and are harmonic conjugates: and
Given , we can recover (up to a constant) using the Cauchy-Riemann equations.
Examples
Example 1:
,
â
â
Example 2:
,
Cauchy-Riemann equations are satisfied.
Modulus of Derivative
Conformality
Polar Form
In polar coordinates :
and
Applications
Jacobian Matrix
The Jacobian of the mapping is:
where . This shows the mapping is a composition of rotation and scaling.