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.
CoulombH2 can be used with CenteredInterval objects. »

Examples

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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:

Asymptotic behavior at 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:

CoulombH2 can be used with CenteredInterval objects:

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 parts of CoulombH2:

Plot the real part of :

Plot the imaginary part of :

Function Properties (6)

Function domain of CoulombH2:

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:

TraditionalForm formatting:

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 a branch cut at , while the built-in CoulombH2 has a branch cut at :

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