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CoulombF
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CoulombF

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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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-e10bo7
Out[1]=1

Evaluate to arbitrary precision:

In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-biuyy
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-r4lqt
Out[1]=1

Complex plot:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-hrb7dz
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-beuztj
Out[1]=1

Asymptotic behavior for large radius:

In[2]:=2
✖
https://wolfram.com/xid/0dcz82e3qtfe-ixik9m
Out[2]=2

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

Numerical Evaluation  (5)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-ccfgcz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-h99tsy
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-cdczwe
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dcz82e3qtfe-5iw71
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dcz82e3qtfe-mst7ur
Out[3]=3

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-biiytw
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-ou7s
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-dd95ec
Out[2]=2

CoulombF can be used with Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-x7cms
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dcz82e3qtfe-kmsryh
Out[2]=2

Specific Values  (3)

Limiting value at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-fxpyz
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-cjepcr
Out[1]=1

Find the first positive zero of CoulombF:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-m7pk
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-mbmovs
Out[2]=2

Visualization  (3)

Plot the CoulombF function:

In[4]:=4
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https://wolfram.com/xid/0dcz82e3qtfe-ecj8m7
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-6y4wh
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-x4ex0
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-epb4bn
Out[1]=1

Function Properties  (7)

Function domain of CoulombF:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-cl7ele
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-de3irc
Out[2]=2

CoulombF is an analytic function of η:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-h5x4l2
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-gi38d7
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-ctca0g
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-84dui
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-mdtl3h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-mn5jws
Out[2]=2

CoulombF is neither convex nor concave:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-6k0d4

Series Expansions  (1)

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

In[10]:=10
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https://wolfram.com/xid/0dcz82e3qtfe-ewr1h8
Out[10]=10
In[9]:=9
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https://wolfram.com/xid/0dcz82e3qtfe-fne0md
Out[9]=9

Plots of the first three approximations for CoulombF around :

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-binhar
Out[2]=2

Function Representations  (1)

Representation through other Coulomb functions:

In[5]:=5
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https://wolfram.com/xid/0dcz82e3qtfe-gfgkhv
Out[5]=5

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

Solve the Coulomb wave equation:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-jvq87f
Out[1]=1

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 :

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-b26nzw

Verify that the wavefunction satisfies the Schrödinger equation for specific values of the energy and separation:

In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-isged
Out[2]=2

Plot the wavefunction:

In[3]:=3
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https://wolfram.com/xid/0dcz82e3qtfe-bczmfx
Out[3]=3

Construct a WKB approximation of CoulombF:

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-eugoe1

Compare the WKB approximation with the actual function:

In[2]:=2
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https://wolfram.com/xid/0dcz82e3qtfe-haict
Out[2]=2

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

CoulombF is a linear combination of CoulombH1 and CoulombH2:

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-bwtlm
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0dcz82e3qtfe-ghsxrr
In[2]:=2
✖
https://wolfram.com/xid/0dcz82e3qtfe-lehkya
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0dcz82e3qtfe-2wxcj
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0dcz82e3qtfe-e9fsp9
Out[1]=1
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 ]}

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@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 ]}

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@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 ]}