CoulombH2

CoulombH2[l,η,r]

gives the incoming irregular Coulomb wavefunction .

Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• CoulombH2[l,η,r] is a solution of the ordinary differential equation .
• CoulombH2[l,η,r] is proportional to for large .
• CoulombH2[l,η,r] has a regular singularity at .
• CoulombH2 has a branch cut discontinuity in the complex plane running from to .
• For certain special arguments, CoulombH2 automatically evaluates to exact values.
• CoulombH2 can be evaluated to arbitrary numerical precision.
• CoulombH2 automatically threads over lists.

Examples

open allclose all

Basic Examples(5)

Evaluate numerically:

Evaluate to arbitrary precision:

CoulombH2 is a linear combination of the CoulombG and CoulombF functions:

Complex plot:

Symbolic evaluation for special parameters:

Scope(18)

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, CoulombH2 reduces to a spherical Hankel function:

Find the first positive zero of the real part of CoulombH2:

Visualization(2)

Plot the real and imaginary values of the CoulombH2 function:

Plot the real part of CoulombH2[2,0,x+I y]:

Plot the imaginary part of CoulombH2[2,0,x+I y]:

Function Properties(7)

Function domain of CoulombH2:

CoulombH2 is an analytic function of η:

CoulombH2[2,0,x] is not injective over complexes:

CoulombH2[2,0,x] is neither non-negative nor non-positive:

CoulombH2[2,0,x] has both singularities and discontinuities:

CoulombH2 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)

CoulombH2 is proportional to WhittakerW in some region of the complex plane:

However, the stated definition has branch cut at , while the built-in CoulombH2 has branch cut at :