# HeunBPrime

HeunBPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunB function.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• HeunBPrime belongs to the Heun class of functions.
• For certain special arguments, HeunBPrime automatically evaluates to exact values.
• HeunBPrime can be evaluated for arbitrary complex parameters.
• HeunBPrime can be evaluated to arbitrary numerical precision.
• HeunBPrime automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot HeunBPrime:

Series expansion of HeunBPrime:

## Scope(22)

### Numerical Evaluation(8)

Evaluate to high precision:

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

HeunBPrime can take one or more complex number parameters:

HeunBPrime can take complex number arguments:

Finally, HeunBPrime can take all complex number input:

Evaluate HeunBPrime efficiently at high precision:

Lists and matrices:

Compute the elementwise values of an array:

Or compute the matrix HeunBPrime function using MatrixFunction:

### Specific Values(1)

Value of HeunBPrime at origin:

### Visualization(5)

Plot the HeunBPrime function:

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

Plot HeunBPrime as a function of its second parameter :

Plot HeunBPrime as a function of and :

Plot the family of HeunBPrime functions for different accessory parameter :

### Differentiation(1)

The derivatives of HeunBPrime are calculated using the HeunB function:

### Integration(3)

Integral of HeunBPrime gives back HeunB:

Definite numerical integral of HeunBPrime:

More integrals with HeunBPrime:

### Series Expansions(4)

Taylor expansion for HeunBPrime at regular singular origin:

Coefficient of the first term in the series expansion of HeunBPrime at :

Plots of the first three approximations for HeunBPrime around :

Series expansion for HeunBPrime at any ordinary complex point:

## Applications(1)

Use the HeunBPrime function to calculate the derivatives of HeunB:

## Properties & Relations(3)

HeunBPrime is analytic at the origin:

HeunBPrime can be calculated at any finite complex :

HeunBPrime is the derivative of HeunB:

## Possible Issues(1)

HeunBPrime diverges for big arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_heunbprime, author="Wolfram Research", title="{HeunBPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunBPrime.html}", note=[Accessed: 13-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_heunbprime, organization={Wolfram Research}, title={HeunBPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunBPrime.html}, note=[Accessed: 13-September-2024 ]}