Cauchy–Riemann equations

Cauchy-Riemann

Contours of holomorphic function

Cauchy–Riemann equations are a set of two partial differential equations that, in complex analysis, establish the conditions under which a complex function is differentiable in the complex sense. Named after Augustin-Louis Cauchy and Bernhard Riemann, these equations are fundamental in the field of complex analysis for they provide a direct method to determine the analyticity of functions in the complex plane.

Definition

Consider a complex function \(f(z)\) where \(z = x + iy\) and \(f(z) = u(x, y) + iv(x, y)\), with \(u\) and \(v\) being real-valued functions of two real variables \(x\) and \(y\), and \(i\) is the imaginary unit. The Cauchy–Riemann equations are:

\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]

These equations must hold for \(f(z)\) to be differentiable at a point in the complex plane.

Implications

The satisfaction of the Cauchy–Riemann equations at a point implies that the complex function \(f(z)\) is holomorphic at that point, meaning it is complex differentiable at that point and in a neighborhood around it. Holomorphic functions have several important properties, including being infinitely differentiable and conformal (angle-preserving) at points where the derivative is non-zero.

Applications

The Cauchy–Riemann equations are used in various applications within mathematics and physics, including:

- Determining the analyticity of complex functions. - Solving Laplace's equation in two dimensions, as the real and imaginary parts of any analytic function satisfy Laplace's equation, making them harmonic functions. - In fluid dynamics, where potential flows can be modeled using complex potential functions that satisfy the Cauchy–Riemann equations.

Examples

A classic example of a function satisfying the Cauchy–Riemann equations is the complex exponential function \(f(z) = e^z\), which is analytic everywhere in the complex plane.

See also

• Complex analysis
• Holomorphic function
• Laplace's equation
• Harmonic function