gives the tri-confluent Heun function.


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


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Basic Examples  (3)

Evaluate numerically:

Plot the HeunT function:

Series expansion of HeunT:

Scope  (23)

Numerical Evaluation  (7)

Evaluate to high precision:

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

HeunT can take one or more complex number parameters:

HeunT can take complex number arguments:

Finally, HeunT can take all complex number input:

Evaluate HeunT efficiently at high precision:

Lists and matrices:

Specific Values  (2)

Value of HeunT at origin:

Calculate HeunT for arbitrary arguments:

Visualization  (5)

Plot the HeunT function:

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

Plot HeunT as a function of its second parameter :

Plot HeunT as a function of and :

Plot the family of HeunT functions for different accessory parameter :

Differentiation  (2)

First -derivative of HeunT is HeunTPrime:

Higher derivatives of HeunT are calculated using HeunTPrime:

Integration  (3)

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

Definite numerical integral of HeunT:

More integrals with HeunT:

Series Expansions  (4)

Taylor expansion for HeunT at origin:

Coefficient of the third term in the series expansion of HeunT at :

Plot the first three approximations for HeunT around :

Series expansion for HeunT at any ordinary complex point:

Applications  (5)

Solve the tri-confluent Heun differential equation using DSolve:

Plot the solution:

Solve the initial value problem for the tri-confluent Heun differential equation:

Plot the solution for different values of the accessory parameter q:

Directly solve the tri-confluent Heun differential equation:

The quartic potential for the 1D Schrödinger equation:

Solve this general potential in terms of HeunT functions:

HeunT solves the Airy equation for a specific set of parameters:

Properties & Relations  (3)

HeunT is analytic at the origin:

HeunT can be calculated at any finite complex :

The derivative of HeunT is HeunTPrime:

Possible Issues  (1)

HeunT calculations might take time for big arguments:

Neat Examples  (1)

The classical anharmonic oscillator equation is solved in terms of HeunT:

Simulate the anharmonic oscillator dynamics:

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


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


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


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


@misc{reference.wolfram_2022_heunt, author="Wolfram Research", title="{HeunT}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunT.html}", note=[Accessed: 07-February-2023 ]}


@online{reference.wolfram_2022_heunt, organization={Wolfram Research}, title={HeunT}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunT.html}, note=[Accessed: 07-February-2023 ]}