WOLFRAM

CoulombF[l,η,r]

gives the regular Coulomb wavefunction TemplateBox[{l, eta, r}, CoulombF].

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)Summary of the most common use cases

Evaluate numerically:

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Evaluate to arbitrary precision:

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

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Complex plot:

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Series expansion at the origin:

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Asymptotic behavior for large radius:

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Scope  (20)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate numerically:

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Evaluate to high precision:

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

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Complex number inputs:

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Evaluate efficiently at high precision:

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CoulombF can be used with Interval and CenteredInterval objects:

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Specific Values  (3)

Limiting value at the origin:

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

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Find the first positive zero of CoulombF:

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Visualization  (3)

Plot the CoulombF function:

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

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

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Polar plot with r=TemplateBox[{2, 0, {k,  , phi}}, CoulombF]:

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Function Properties  (7)

Function domain of CoulombF:

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CoulombF is an analytic function of η:

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

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

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

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CoulombF is neither convex nor concave:

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TraditionalForm formatting:

Series Expansions  (1)

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

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Plots of the first three approximations for CoulombF around :

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Function Representations  (1)

Representation through other Coulomb functions:

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Applications  (3)Sample problems that can be solved with this function

Solve the Coulomb wave equation:

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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:

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Plot the wavefunction:

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Construct a WKB approximation of CoulombF:

Compare the WKB approximation with the actual function:

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Properties & Relations  (2)Properties of the function, and connections to other functions

CoulombF is a linear combination of CoulombH1 and CoulombH2:

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

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

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Neat Examples  (1)Surprising or curious use cases

Plot the Riemann surface of TemplateBox[{{3, /, 5}, {{-, 1}, /, 2}, rho}, CoulombF]:

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

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_coulombf, author="Wolfram Research", title="{CoulombF}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CoulombF.html}", note=[Accessed: 11-July-2025 ]}

@misc{reference.wolfram_2025_coulombf, author="Wolfram Research", title="{CoulombF}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CoulombF.html}", note=[Accessed: 11-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_coulombf, organization={Wolfram Research}, title={CoulombF}, year={2023}, url={https://reference.wolfram.com/language/ref/CoulombF.html}, note=[Accessed: 11-July-2025 ]}

@online{reference.wolfram_2025_coulombf, organization={Wolfram Research}, title={CoulombF}, year={2023}, url={https://reference.wolfram.com/language/ref/CoulombF.html}, note=[Accessed: 11-July-2025 ]}