gives the bi-confluent Heun function.


  • HeunB belongs to the Heun class of functions and occurs in quantum mechanics, mathematical physics and applications.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • HeunB[q,α,γ,δ,ϵ,z] satisfies the bi-confluent Heun differential equation .
  • The HeunB function is the regular solution of the bi-confluent Heun equation that satisfies the condition HeunB[q,α,γ,δ,ϵ,0]=1.
  • For certain special arguments, HeunB automatically evaluates to exact values.
  • HeunB can be evaluated for arbitrary complex parameters.
  • HeunB can be evaluated to arbitrary numerical precision.
  • HeunB automatically threads over lists.


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Basic Examples  (3)

Evaluate numerically:

Plot the HeunB function:

Series expansion of HeunB:

Scope  (23)

Numerical Evaluation  (7)

Evaluate to high precision:

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

HeunB can take one or more complex number parameters:

HeunB can take complex number arguments:

Finally, HeunB can take all complex number input:

Evaluate HeunB efficiently at high precision:

Lists and matrices:

Specific Values  (1)

Value of HeunB at origin:

Visualization  (5)

Plot the HeunB function:

Plot the absolute value of the HeunB function for complex parameters:

Plot HeunB as a function of its second parameter :

Plot HeunB as a function of and :

Plot the family of HeunB functions for different accessory parameter :

Function Properties  (1)

HeunB can be simplified to Hypergeometric1F1 function in the following case:

Differentiation  (2)

The -derivative of HeunB is HeunBPrime:

Higher derivatives of HeunB are calculated using HeunBPrime:

Integration  (3)

Indefinite integrals of HeunB are not expressed in elementary or other special functions:

Definite numerical integral of HeunB:

More integrals with HeunB:

Series Expansions  (4)

Taylor expansion for HeunB at regular singular origin:

Coefficient of the second term in the series expansion of HeunB at :

Plot the first three approximations for HeunB around :

Series expansion for HeunB at any ordinary complex point:

Applications  (3)

Solve the bi-confluent Heun differential equation using DSolve:

Plot the solution:

Directly solve the bi-confluent Heun differential equation:

Solve the class of confinement potentials for the radial Schrödinger equation in terms of HeunB functions:

Plot the potential for arbitrary parameters:

This general potential is solved in terms of HeunB functions:

Properties & Relations  (3)

HeunB is analytic at the origin:

HeunB can be calculated at any finite complex :

The derivative of HeunB is HeunBPrime:

Possible Issues  (1)

HeunB diverges for big arguments:

Neat Examples  (2)

Create a table of some special cases for HeunB :

The quantum-mechanical doubly anharmonic oscillator potential is:

Plot the potential:

The general solution of the Schrödinger equation is written in terms of HeunB functions:

Verify this solution by direct substitution:

Wolfram Research (2020), HeunB, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunB.html.


Wolfram Research (2020), HeunB, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunB.html.


Wolfram Language. 2020. "HeunB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HeunB.html.


Wolfram Language. (2020). HeunB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeunB.html


@misc{reference.wolfram_2022_heunb, author="Wolfram Research", title="{HeunB}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunB.html}", note=[Accessed: 11-August-2022 ]}


@online{reference.wolfram_2022_heunb, organization={Wolfram Research}, title={HeunB}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunB.html}, note=[Accessed: 11-August-2022 ]}