Wirtinger

Wirtinger refers to Wilhelm Wirtinger, a mathematician whose work in complex analysis, topology, and geometry led to several concepts that bear his name. The most widely used are Wirtinger derivatives, the Wirtinger inequality, and the Wirtinger presentation in knot theory. Together they illustrate the influence of his ideas across different areas of mathematics.

Wirtinger derivatives are a formalism for functions of a complex variable using the variables z = x + iy and its conjugate z̄ = x − iy. The derivatives are defined by ∂/∂z = 1/2(∂/∂x − i∂/∂y) and ∂/∂z̄ = 1/2(∂/∂x + i∂/∂y). These operators simplify the expression of the Cauchy–Riemann equations, give a compact description of holomorphicity (a function f is holomorphic precisely when ∂f/∂z̄ = 0), and are central in several complex variables, complex geometry, and partial differential equations through the d̄-operator.

The Wirtinger inequality is a result in Fourier analysis and functional analysis. It states that for a 2π-periodic function f with zero mean, the L2-norm of f is bounded by the L2-norm of its derivative: ∫0^{2π} f^2 ≤ ∫0^{2π} (f')^2, with equality achieved by certain sine or cosine functions. The inequality serves as a special case of Poincaré-type estimates on the circle and appears in approximation theory and spectral analysis.

In knot theory, the Wirtinger presentation gives a presentation of a knot group from a diagram. It assigns generators to the arcs and relations at crossings, producing a group that is the fundamental group of the knot complement. This presentation is named after Wirtinger and remains a standard computational tool in studying knot invariants.

The name Wirtinger thus marks several distinct ideas in mathematics, all linked to the contributions of Wilhelm Wirtinger.