Cauchy-Riemann Equations

★★★★☆Undergraduate

📖Definition

Cauchy-Riemann equations are necessary and sufficient conditions for a complex function to be differentiable (analytic). They relate partial derivatives of real and imaginary parts.

📐Formulas

(∂ u)/(∂ x) = (∂ v)/(∂ y)

First Cauchy-Riemann equation

(∂ u)/(∂ y) = -(∂ v)/(∂ x)

Second Cauchy-Riemann equation

f'(z) = (∂ u)/(∂ x) + i(∂ v)/(∂ x)

Complex derivative

✏️Examples

예제 1

Verify f(z) = z² is analytic using Cauchy-Riemann.

예제 2

Show f(z) = |z|² is not analytic.

📜History

Discovered by: Augustin-Louis Cauchy, Bernhard Riemann (1851)

Cauchy first derived them; Riemann generalized, forming basis of complex analysis.

⚡Applications

Fluid Mechanics

Incompressible flow

Electromagnetism

Electrostatic potential

Elasticity

2D stress analysis

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