The Helmholtz equation arises in many contexts in the attempt to give a mathematical explanation of the physical world. These range from Alan Turing's explanation of animal coat patterns to Schrödinger's time-independent equation in quantum theory.

The Helmholtz equation per se is

∇²φ = −k²φ

where k is a constant. The Generalized Helmholtz equation is that equation with k a function of the independent variable.

In one dimension the Helmholtz equation just has the sinusoidal solution of φ(x) = A·sin(kx)+B·cos(kx). In one dimension the Generalized Helmholtz equation has a sinusoidal-like solution of varying amplitude and wavelength.

• The Helmholtz Equation and Its Solution
• A Property of Generalized Helmholtz Equations of Dimension One or More
• A Property of Solutions to Generalized Helmholtz Equations and Its Implication for the Relationship of Classical and Quantum Theoretic Probability Density Functions
• A Property of Solutions to a Generalized Helmholtz Equation
• The Solution to a Generalized Helmholtz Equation of One Dimension
• The Spatial Average of the Probability Density Function Derived from the Solution to a Generalized Helmholtz Equation of One Dimension
• The Spatial Average of the Probability Density Function Derived from the Solution to a Generalized Helmholtz Equation of One Dimension
• The Solution to the Generalized Helmholtz Equation in Matrix Form
• A Transformation of a Generalized Helmholtz Equation
• The relative extrema of the solution to a generalized Helmholtz equation
• The Solution of a Generalized Helmholtz Equation of Two Dimensions
• A property of the Solution of a Generalized Helmholtz Equation of Any Dimension
• An Analytical Solution to the Generalized Helmholtz Equation of One Dimension