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CoulombG[l,η,r]

gives the irregular Coulomb wavefunction TemplateBox[{l, eta, r}, CoulombG].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Series Expansions  
Function Representations  
Applications  
Properties & Relations  
See Also
Related Guides
History
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CoulombG

CoulombG[l,η,r]

gives the irregular Coulomb wavefunction TemplateBox[{l, eta, r}, CoulombG].

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.
  • CoulombG can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Wolfram Language code: CoulombG[3, 0.12, 7.]
Wolfram Language code: CoulombG[3, 0.12`50, 3`50]

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

Wolfram Language code: Plot[{CoulombG[0, 0.3, r], CoulombG[0, -0.3, r]}, {r, 0, 10}]

Complex plot:

Wolfram Language code: ComplexPlot[CoulombG[0, 2, z], {z, -3 - 3I, 3 + 3I}]

Series expansion at the origin:

Wolfram Language code: Series[CoulombG[0, 2, r], {r, 0, 1}]//TraditionalForm

Asymptotic behavior for large radius:

Wolfram Language code: Asymptotic[CoulombG[ℓ, η, r], r -> ∞]

Scope  (18)

Numerical Evaluation  (5)

Evaluate numerically:

Wolfram Language code: N[CoulombG[1 / 2, 7, 1]]
Wolfram Language code: N[CoulombG[1 / 2, -7, 1]]

Evaluate to high precision:

Wolfram Language code: N[CoulombG[1 / 2, 7, 1], 50]

The precision of the output tracks the precision of the input:

Wolfram Language code: CoulombG[-2 / 5, -3, 1.23456789101112131415]
Wolfram Language code: CoulombG[-2 / 5, -3, 1.2345678910111213141516171819202122]

Complex number inputs:

Wolfram Language code: CoulombG[2 / 3, 0.7 + 0.1 I, 3.2]

Evaluate efficiently at high precision:

Wolfram Language code: CoulombG[11, -3, 70`100]//NumberForm[#, 20]&//Timing
Wolfram Language code: CoulombG[11, -3, 70`1000]//NumberForm[#, 20]&//Timing

CoulombG can be used with Interval and CenteredInterval objects:

Wolfram Language code: CoulombG[Interval[{0.234, 0.235}], Interval[{0.345, 0.346}], Interval[{0.456, 0.457}]]
Wolfram Language code: CoulombG[CenteredInterval[3 / 4, 10 ^ -6], CenteredInterval[5 / 6, 10 ^ -6], CenteredInterval[7 / 8, 10 ^ -6]]

Specific Values  (2)

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

Wolfram Language code: CoulombG[n, 0, r]

Find the first positive zero of CoulombG:

Wolfram Language code: rzero = r /. FindRoot[CoulombG[0, 3, r], {r, 9}]
Wolfram Language code: Plot[CoulombG[0, 3, r], {r, 0, 20}, Epilog -> Style[Point[{rzero, CoulombG[0, 3, rzero]}], PointSize[Large], Red]]

Visualization  (2)

Plot the CoulombG function:

Wolfram Language code: Plot[CoulombG[2, 0, x], {x, 0, 4π}]

Plot the real part of TemplateBox[{2, 0, z}, CoulombG]:

Wolfram Language code: ComplexContourPlot[Re[CoulombG[2, 0, z]], {z, -2π - π I, 2π + π I}, IconizedObject[«PlotOptions»]]

Plot the imaginary part of TemplateBox[{2, 0, z}, CoulombG]:

Wolfram Language code: ComplexContourPlot[Im[CoulombG[2, 0, z]], {z, -2π - π I, 2π + π I}, IconizedObject[«PlotOptions»]]

Function Properties  (7)

Function domain of CoulombG:

Wolfram Language code: FunctionDomain[CoulombG[n, η, x], x]
Wolfram Language code: FunctionDomain[CoulombG[n, η, z], z, Complexes]

CoulombG is an analytic function of η:

Wolfram Language code: FunctionAnalytic[CoulombG[n, η, x], η]

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

Wolfram Language code: FunctionInjective[CoulombG[2, 0, x], x]
Wolfram Language code: Plot[{CoulombG[2, 0, x], 1 / 2}, {x, -4π, 4π}]

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

Wolfram Language code: FunctionSign[CoulombG[2, 0, x], x]

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

Wolfram Language code: FunctionSingularities[CoulombG[2, 0, x], x]
Wolfram Language code: FunctionDiscontinuities[CoulombG[2, 0, x], x]

CoulombG is neither convex nor concave:

Wolfram Language code: FunctionConvexity[CoulombG[2, 0, x], x]

TraditionalForm formatting:

Wolfram Language code: CoulombG[n, η, z]//TraditionalForm

Series Expansions  (1)

Find the Taylor expansion using Series at zero and at infinity:

Wolfram Language code: Series[CoulombG[1, 2, x], {x, 0, 2}]//FullSimplify
Wolfram Language code: Series[CoulombG[1, 2, x], {x, ∞, 3}]//FullSimplify

Plots of the first three approximations for CoulombG around :

Wolfram Language code: terms = Normal@Table[Series[CoulombG[1, 0, x], {x, 0, m}], {m, 1, 5, 2}]; Plot[{CoulombG[1, 0, x], terms}, {x, -2π, 2π}, PlotRange -> {-10, 10}]

Function Representations  (1)

Representation through other Coulomb functions:

Wolfram Language code: (CoulombH2[ℓ, η, r] + CoulombH1[ℓ, η, r]) / 2//FullSimplify

Applications  (2)

Solve the Coulomb wave equation:

Wolfram Language code: DSolve[{y''[r] + (1 - (2η/r) - (ℓ(ℓ + 1)/r^2))y[r] == 0}, y[r], r]

Construct a WKB approximation of CoulombG:

Wolfram Language code: wkbCoulombG[el_, eta_, r_ ? NumberQ] := Module[{rhoT = eta + √(eta^2 + el(el + 1)), a, phi}, CompoundExpression[...] ]

Compare the WKB approximation with the actual function:

Wolfram Language code: Plot[{CoulombG[3, 17, r], wkbCoulombG[3, 17, r]}, {r, 20, 100}, PlotStyle -> {Thick, Dashed}]

Properties & Relations  (1)

CoulombG is a linear combination of CoulombH1 and CoulombH2:

Wolfram Language code: (CoulombH1[ℓ, η, r] + CoulombH2[ℓ, η, r]) / 2//FullSimplify
Wolfram Research (2021), CoulombG, Wolfram Language function, https://reference.wolfram.com/language/ref/CoulombG.html (updated 2023).

Text

Wolfram Research (2021), CoulombG, Wolfram Language function, https://reference.wolfram.com/language/ref/CoulombG.html (updated 2023).

CMS

Wolfram Language. 2021. "CoulombG." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CoulombG.html.

APA

Wolfram Language. (2021). CoulombG. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoulombG.html

BibTeX

@misc{reference.wolfram_2026_coulombg, author="Wolfram Research", title="{CoulombG}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CoulombG.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_coulombg, organization={Wolfram Research}, title={CoulombG}, year={2023}, url={https://reference.wolfram.com/language/ref/CoulombG.html}, note=[Accessed: 13-June-2026]}