Complex Analysis explores the properties and applications of complex numbers and functions, emphasizing their geometric interpretations and analytical techniques.
The Cauchy-Riemann equations provide the conditions that a complex function must satisfy to be analytic.
If \( f(z) = u(x, y) + iv(x, y) \), where \( z = x + iy \), then \( f \) is analytic at a point if the partial derivatives exist and satisfy: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]
These equations ensure that the function preserves angles and shapes locally—a property called conformality.
The Cauchy-Riemann equations are a quick way to check if a function is analytic and to discover the magical properties of complex functions.
For \( f(z) = z^2 \), the real and imaginary parts satisfy the Cauchy-Riemann equations everywhere.
For \( f(z) = \overline{z} \), the equations fail, so it's not analytic.
The Cauchy-Riemann equations are the test for complex differentiability and analyticity.