CoulombF
✖
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 allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0dcz82e3qtfe-e10bo7

Evaluate to arbitrary precision:

https://wolfram.com/xid/0dcz82e3qtfe-biuyy

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

https://wolfram.com/xid/0dcz82e3qtfe-r4lqt


https://wolfram.com/xid/0dcz82e3qtfe-hrb7dz

Series expansion at the origin:

https://wolfram.com/xid/0dcz82e3qtfe-beuztj

Asymptotic behavior for large radius:

https://wolfram.com/xid/0dcz82e3qtfe-ixik9m

Scope (20)Survey of the scope of standard use cases
Numerical Evaluation (5)

https://wolfram.com/xid/0dcz82e3qtfe-ccfgcz


https://wolfram.com/xid/0dcz82e3qtfe-h99tsy


https://wolfram.com/xid/0dcz82e3qtfe-cdczwe

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

https://wolfram.com/xid/0dcz82e3qtfe-5iw71


https://wolfram.com/xid/0dcz82e3qtfe-mst7ur


https://wolfram.com/xid/0dcz82e3qtfe-biiytw

Evaluate efficiently at high precision:

https://wolfram.com/xid/0dcz82e3qtfe-ou7s


https://wolfram.com/xid/0dcz82e3qtfe-dd95ec

CoulombF can be used with Interval and CenteredInterval objects:

https://wolfram.com/xid/0dcz82e3qtfe-x7cms


https://wolfram.com/xid/0dcz82e3qtfe-kmsryh

Specific Values (3)

https://wolfram.com/xid/0dcz82e3qtfe-fxpyz

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

https://wolfram.com/xid/0dcz82e3qtfe-cjepcr

Find the first positive zero of CoulombF:

https://wolfram.com/xid/0dcz82e3qtfe-m7pk


https://wolfram.com/xid/0dcz82e3qtfe-mbmovs

Visualization (3)
Plot the CoulombF function:

https://wolfram.com/xid/0dcz82e3qtfe-ecj8m7


https://wolfram.com/xid/0dcz82e3qtfe-6y4wh


https://wolfram.com/xid/0dcz82e3qtfe-x4ex0


https://wolfram.com/xid/0dcz82e3qtfe-epb4bn

Function Properties (7)
Function domain of CoulombF:

https://wolfram.com/xid/0dcz82e3qtfe-cl7ele


https://wolfram.com/xid/0dcz82e3qtfe-de3irc

CoulombF is an analytic function of η:

https://wolfram.com/xid/0dcz82e3qtfe-h5x4l2

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

https://wolfram.com/xid/0dcz82e3qtfe-gi38d7


https://wolfram.com/xid/0dcz82e3qtfe-ctca0g

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

https://wolfram.com/xid/0dcz82e3qtfe-84dui

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

https://wolfram.com/xid/0dcz82e3qtfe-mdtl3h


https://wolfram.com/xid/0dcz82e3qtfe-mn5jws

CoulombF is neither convex nor concave:

https://wolfram.com/xid/0dcz82e3qtfe-kdss3

TraditionalForm formatting:

https://wolfram.com/xid/0dcz82e3qtfe-6k0d4

Series Expansions (1)
Find the Taylor expansion using Series at zero and at infinity:

https://wolfram.com/xid/0dcz82e3qtfe-ewr1h8


https://wolfram.com/xid/0dcz82e3qtfe-fne0md

Plots of the first three approximations for CoulombF around :

https://wolfram.com/xid/0dcz82e3qtfe-binhar

Applications (3)Sample problems that can be solved with this function
Solve the Coulomb wave equation:

https://wolfram.com/xid/0dcz82e3qtfe-jvq87f

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
:

https://wolfram.com/xid/0dcz82e3qtfe-b26nzw
Verify that the wavefunction satisfies the Schrödinger equation for specific values of the energy and separation:

https://wolfram.com/xid/0dcz82e3qtfe-isged


https://wolfram.com/xid/0dcz82e3qtfe-bczmfx

Construct a WKB approximation of CoulombF:

https://wolfram.com/xid/0dcz82e3qtfe-eugoe1
Compare the WKB approximation with the actual function:

https://wolfram.com/xid/0dcz82e3qtfe-haict

Properties & Relations (2)Properties of the function, and connections to other functions
CoulombF is a linear combination of CoulombH1 and CoulombH2:

https://wolfram.com/xid/0dcz82e3qtfe-bwtlm

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

https://wolfram.com/xid/0dcz82e3qtfe-ghsxrr

https://wolfram.com/xid/0dcz82e3qtfe-lehkya

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

https://wolfram.com/xid/0dcz82e3qtfe-2wxcj

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
]}
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
]}