CoulombG

CoulombG[l,η,r]

gives the irregular Coulomb wavefunction .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
CoulombG[l,η,r] is a solution of the ordinary differential equation .
CoulombG[l,η,r] tends to for large and some phase shift .
CoulombG[l,η,r] has a regular singularity at .
CoulombG has a branch cut discontinuity in the complex plane running from to .
For certain special arguments, CoulombG automatically evaluates to exact values.
CoulombG can be evaluated to arbitrary numerical precision.
CoulombG automatically threads over lists.
CoulombG can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (4)

Evaluate numerically:

Plot the Coulomb wavefunction for repulsive () and attractive () interactions:

Complex plot:

Series expansion at the origin:

Asymptotic behavior for large radius:

Scope (18)

Numerical Evaluation (5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

CoulombG can be used with Interval and CenteredInterval objects:

Specific Values (2)

For a zero value of the parameter η, CoulombG reduces to a spherical Bessel function:

Find the first positive zero of CoulombG:

Visualization (2)

Plot the CoulombG function:

Plot the real part of :

Plot the imaginary part of :

Function Properties (7)

Function domain of CoulombG:

CoulombG is an analytic function of η:

CoulombG[2,0,x] is not injective:

CoulombG[2,0,x] is neither non-negative nor non-positive:

CoulombG[2,0,x] has both singularities and discontinuities at zero:

CoulombG is neither convex nor concave:

TraditionalForm formatting:

Series Expansions (1)

Find the Taylor expansion using Series at zero and at infinity:

Plots of the first three approximations for CoulombG around :

Function Representations (1)

Representation through other Coulomb functions:

Applications (2)

Solve the Coulomb wave equation:

Construct a WKB approximation of CoulombG:

Compare the WKB approximation with the actual function:

Properties & Relations (1)

CoulombG is a linear combination of CoulombH1 and CoulombH2:

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