# HeunD

HeunD[q,α,γ,δ,ϵ,z]

gives the double-confluent Heun function.

# Details

• HeunD 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.
• HeunD[q,α,γ,δ,ϵ,z] satisfies the double-confluent Heun differential equation .
• The HeunD function is the power-series solution of the double-confluent Heun equation that satisfies the conditions and .
• For certain special arguments, HeunD automatically evaluates to exact values.
• HeunD can be evaluated for arbitrary complex parameters.
• HeunD can be evaluated to arbitrary numerical precision.
• HeunD automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the double-confluent Heun function:

Series expansion of HeunD:

## Scope(24)

### Numerical Evaluation(8)

Evaluate to high precision:

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

HeunD can take one or more complex number parameters:

HeunD can take complex number arguments:

Finally, HeunD can take all complex number input:

Evaluate HeunD efficiently at high precision:

Lists and matrices:

Evaluate HeunD for points on the real negative axis, bypassing irregular singular origin:

### Specific Values(2)

Value of HeunD at unit point:

Value of HeunD at irregular singular origin is undetermined:

### Visualization(5)

Plot the HeunD function:

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

Plot HeunD as a function of its second parameter :

Plot HeunD as a function of and :

Plot the family of HeunD functions for different accessory parameter :

### Differentiation(2)

The -derivative of HeunD is HeunDPrime:

Higher derivatives of HeunD are calculated using HeunDPrime:

### Integration(3)

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

Definite numerical integral of HeunD:

More integrals with HeunD:

### Series Expansions(4)

Taylor expansion for HeunD at regular point :

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

Plot the first three approximations for HeunD around :

Series expansion for HeunD at any ordinary complex point:

## Applications(3)

Solve the double-confluent Heun differential equation using DSolve:

Plot the solution:

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

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

Directly solve the double-confluent Heun differential equation:

## Properties & Relations(3)

HeunD is analytic at the point :

Origin is a singular point of the HeunD function:

Except for this singular point, HeunD can be calculated at any finite complex :

The derivative of HeunD is HeunDPrime:

## Possible Issues(1)

HeunD diverges for big arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_heund, author="Wolfram Research", title="{HeunD}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunD.html}", note=[Accessed: 20-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_heund, organization={Wolfram Research}, title={HeunD}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunD.html}, note=[Accessed: 20-April-2024 ]}