# CoulombH1

CoulombH1[l,η,r]

gives the outgoing irregular Coulomb wavefunction .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• CoulombH1[,η,r] is a solution of the ordinary differential equation .
• CoulombH1[l,η,r] is proportional to for large .
• CoulombH1[l,η,r] has a regular singularity at .
• CoulombH1 has a branch cut discontinuity in the complex plane running from to .
• For certain special arguments, CoulombH1 automatically evaluates to exact values.
• CoulombH1 can be evaluated to arbitrary numerical precision.
• CoulombH1 automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Evaluate to arbitrary precision:

CoulombH1 is a linear combination of the CoulombG and CoulombF functions:

Complex plot:

Symbolic evaluation for special parameters:

## 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(3)

Limiting value at the origin:

For a zero value of the parameter η, CoulombH1 reduces to a spherical Hankel function:

Find the first positive zero of the real part of CoulombH1:

### Visualization(2)

Plot the real and imaginary values of CoulombH1 function:

Plot the real part of CoulombH1[2,0,x+I y]:

Plot the imaginary part of CoulombH1[2,0,x+I y]:

### Function Properties(6)

Function domain of CoulombH1:

CoulombH1[2,0,x] is not injective over complexes:

CoulombH1[2,0,x] is neither non-negative nor non-positive:

CoulombH1[2,0,x] has both singularities and discontinuities:

CoulombH1 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 CoulombH1 around :

### Function Representations(1)

Relations with other Coulomb functions:

## Properties & Relations(1)

CoulombH1 is proportional to WhittakerW in some region of the complex plane:

However, the stated definition has branch cut at , while the built-in CoulombH1 has branch cut at :