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


  • 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. »


open allclose all

Basic Examples  (4)

Evaluate numerically:

Evaluate to arbitrary precision:

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

Complex plot:

Series expansion at the origin:

Asymptotic behavior for large radius:

Scope  (20)

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:

CoulombF can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Limiting value at the origin:

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

Find the first positive zero of CoulombF:

Visualization  (3)

Plot the CoulombF function:

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

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

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

Function Properties  (7)

Function domain of CoulombF:

CoulombF is an analytic function of η:

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

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

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

CoulombF 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 CoulombF around :

Function Representations  (1)

Representation through other Coulomb functions:

Applications  (3)

Solve the Coulomb wave equation:

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:

Plot the wavefunction:

Construct a WKB approximation of CoulombF:

Compare the WKB approximation with the actual function:

Properties & Relations  (2)

CoulombF is a linear combination of CoulombH1 and CoulombH2:

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

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

Wolfram Research (2021), CoulombF, Wolfram Language function, (updated 2023).


Wolfram Research (2021), CoulombF, Wolfram Language function, (updated 2023).


Wolfram Language. 2021. "CoulombF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023.


Wolfram Language. (2021). CoulombF. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_coulombf, author="Wolfram Research", title="{CoulombF}", year="2023", howpublished="\url{}", note=[Accessed: 19-July-2024 ]}


@online{reference.wolfram_2024_coulombf, organization={Wolfram Research}, title={CoulombF}, year={2023}, url={}, note=[Accessed: 19-July-2024 ]}