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Studies Concerning the Solution ofthe Generalized Helmholtz Equation |
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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

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

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