# HeunGPrime

HeunGPrime[a,q,α,β,γ,δ,z]

gives the -derivative of the HeunG function.

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the HeunGPrime function:

Series expansion of HeunGPrime:

## Scope(27)

### Numerical Evaluation(9)

Evaluate to high precision:

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

HeunGPrime can take one or more complex number parameters:

HeunGPrime can take complex number arguments:

Finally, HeunGPrime can take all complex number input:

Evaluate HeunGPrime efficiently at high precision:

Lists and matrices:

Evaluate HeunGPrime for points at branch cut to :

Evaluate HeunGPrime for points on a branch cut from to :

### Specific Values(5)

Value of HeunGPrime at origin:

Value of HeunGPrime at regular singular point is indeterminate:

Value of HeunGPrime at regular singular point is indeterminate:

Values of HeunGPrime in "logarithmic" cases, i.e. for nonpositive integer , are not determined:

Value of HeunGPrime is not determined if :

### Visualization(5)

Plot the HeunGPrime function:

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

Plot HeunGPrime as a function of its third parameter :

Plot HeunGPrime as a function of and :

Plot the family of HeunGPrime functions for different accessory parameter :

### Differentiation(1)

The derivatives of HeunGPrime are calculated using the HeunG function:

### Integration(3)

Integral of HeunGPrime gives back HeunG:

Definite numerical integral of HeunGPrime:

More integrals with HeunGPrime:

### Series Expansions(4)

Taylor expansion for HeunGPrime at regular singular origin:

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

Plots of the first three approximations for HeunGPrime around :

Series expansion for HeunGPrime at any ordinary complex point:

## Applications(1)

Use the HeunGPrime function to calculate the derivatives of HeunG:

## Properties & Relations(3)

HeunGPrime is analytic at the origin:

is a singular point of the HeunGPrime function:

is a singular point of the HeunGPrime function:

Except for these two singular points, HeunGPrime can be calculated at any finite complex :

HeunGPrime is the derivative of HeunG:

## Possible Issues(2)

HeunGPrime cannot be evaluated if is a nonpositive integer (so-called logarithmic cases):

HeunGPrime is undefined when :

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_heungprime, author="Wolfram Research", title="{HeunGPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunGPrime.html}", note=[Accessed: 05-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_heungprime, organization={Wolfram Research}, title={HeunGPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunGPrime.html}, note=[Accessed: 05-June-2023 ]}