# HeunT

HeunT[q,α,γ,δ,ϵ,z]

gives the tri-confluent Heun function.

# Details

• 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.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the HeunT function:

Series expansion of HeunT:

## Scope(24)

### Numerical Evaluation(8)

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:

Compute the elementwise values of an array:

Or compute the matrix HeunT function using MatrixFunction:

### 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:

When , HeunT can be expressed in terms of Airy functions:

Verify that the result is a solution of the Airy equation:

## Properties & Relations(4)

HeunT is analytic at the origin:

HeunT can be calculated at any finite complex :

The derivative of HeunT is HeunTPrime:

Use FunctionExpand to expand HeunT into simpler functions:

## 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.

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_heunt, author="Wolfram Research", title="{HeunT}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunT.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_heunt, organization={Wolfram Research}, title={HeunT}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunT.html}, note=[Accessed: 14-September-2024 ]}