# CoulombF

CoulombF[l,η,r]

gives the regular Coulomb wavefunction .

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

# Examples

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## Basic Examples(4)

Evaluate numerically:

Evaluate to arbitrary precision:

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

Complex plot:

Series expansion at the origin:

## Scope(20)

### 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:

CoulombF can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Limiting value at the origin:

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

Find the first positive zero of CoulombF:

### Visualization(3)

Plot the CoulombF function:

Plot the real part of CoulombF[2,0,x+I y]:

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

Polar plot with :

### Function Properties(7)

Function domain of CoulombF:

CoulombF is an analytic function of η:

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

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

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

CoulombF 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 CoulombF around :

### Function Representations(1)

Representation through other Coulomb functions:

## Applications(2)

Wavefunction for the radial Schrödinger equation with Coulomb potential between two point particles with charges and separated by a distance and energy of relative motion :

Verify that the wavefunction satisfies the Schrödinger equation for specific values of the energy and separation:

Construct a WKB approximation of CoulombF:

## Properties & Relations(2)

CoulombF is a linear combination of CoulombH1 and CoulombH2:

CoulombF is related to Hypergeometric1F1Regularized in some region of the complex plane:

However, the stated definition has the branch cut at , while the built-in CoulombF has the branch cut at :