# 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.

# 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:

## Scope(17)

### Numerical Evaluation(4)

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:

### 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 CoulombG[2,0,x+I y] :

Plot the imaginary part of CoulombG[2,0,x+I y] :

### 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:

### 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(1)

Construct a WKB approximation of CoulombG:

## Properties & Relations(1)

CoulombG is a linear combination of CoulombH1 and CoulombH2: