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AdvancedMathCoulomb Method
Computes the regular and irregular Coulomb wave functions and their derivatives.

Namespace:  Meta.Numerics.Functions
Assembly:  Meta.Numerics (in Meta.Numerics.dll) Version: 3.1.0.0 (3.1.0.0)
Syntax
public static SolutionPair Coulomb(
	int L,
	double eta,
	double rho
)

Parameters

L
Type: SystemInt32
The angular momentum number, which must be non-negative.
eta
Type: SystemDouble
The charge parameter, which can be postive or negative.
rho
Type: SystemDouble
The radial distance parameter, which must be non-negative.

Return Value

Type: SolutionPair
The values of F, F', G, and G' for the given parameters.
Exceptions
ExceptionCondition
ArgumentOutOfRangeExceptionL or rho is negative.
Remarks

The Coulomb wave functions are the radial wave functions of a non-relativistic particle in a Coulomb potential.

They satisfy the differential equation:

A repulsive potential is represented by η > 0, an attractive potential by η < 0.

F is oscilatory in the region beyond the classical turning point. In the quantum tunneling region inside the classical turning point, F is exponentially supressed and vanishes at the origin, while G grows exponentially and diverges at the origin.

Many numerical libraries compute Coulomb wave functions in the quantum tunneling region using a WKB approximation, which accurately determine only the first handfull of digits; our library computes Coulomb wave functions even in this computationaly difficult region to nearly full precision -- all but the last 3-4 digits can be trusted.

The irregular Coulomb wave functions GL(η,ρ) are the complementary independent solutions of the same differential equation.

See Also