HeunGPrime
✖
HeunGPrime
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
open allclose allBasic Examples (3)Summary of the most common use cases

https://wolfram.com/xid/0bni4ex43wfqs-3kfjq

Plot the HeunGPrime function:

https://wolfram.com/xid/0bni4ex43wfqs-ftt82q

Series expansion of HeunGPrime:

https://wolfram.com/xid/0bni4ex43wfqs-z8evs7

Scope (28)Survey of the scope of standard use cases
Numerical Evaluation (10)

https://wolfram.com/xid/0bni4ex43wfqs-uqurky

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

https://wolfram.com/xid/0bni4ex43wfqs-lw9h0n

HeunGPrime can take one or more complex number parameters:

https://wolfram.com/xid/0bni4ex43wfqs-64g5bd


https://wolfram.com/xid/0bni4ex43wfqs-ft5oo1

HeunGPrime can take complex number arguments:

https://wolfram.com/xid/0bni4ex43wfqs-hunut5

Finally, HeunGPrime can take all complex number input:

https://wolfram.com/xid/0bni4ex43wfqs-56m4mo

Evaluate HeunGPrime efficiently at high precision:

https://wolfram.com/xid/0bni4ex43wfqs-2c7v5i


https://wolfram.com/xid/0bni4ex43wfqs-yaawua


https://wolfram.com/xid/0bni4ex43wfqs-22a9kq


https://wolfram.com/xid/0bni4ex43wfqs-1knfqv


https://wolfram.com/xid/0bni4ex43wfqs-7yjugj

Evaluate HeunGPrime for points at branch cut to
:

https://wolfram.com/xid/0bni4ex43wfqs-vush9d

Evaluate HeunGPrime for points on a branch cut from to DirectedInfinity[a]:

https://wolfram.com/xid/0bni4ex43wfqs-zso2z1

Compute the elementwise values of an array:

https://wolfram.com/xid/0bni4ex43wfqs-thgd2

Or compute the matrix HeunGPrime function using MatrixFunction:

https://wolfram.com/xid/0bni4ex43wfqs-o5jpo

Specific Values (5)
Value of HeunGPrime at origin:

https://wolfram.com/xid/0bni4ex43wfqs-nuboa

Value of HeunGPrime at regular singular point is indeterminate:

https://wolfram.com/xid/0bni4ex43wfqs-124w4g

Value of HeunGPrime at regular singular point is indeterminate:

https://wolfram.com/xid/0bni4ex43wfqs-6iqn02

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

https://wolfram.com/xid/0bni4ex43wfqs-kqbmlj


https://wolfram.com/xid/0bni4ex43wfqs-jzxssp


https://wolfram.com/xid/0bni4ex43wfqs-i7w0vo

Value of HeunGPrime is not determined if :

https://wolfram.com/xid/0bni4ex43wfqs-n91pyz

Visualization (5)
Plot the HeunGPrime function:

https://wolfram.com/xid/0bni4ex43wfqs-n742f

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

https://wolfram.com/xid/0bni4ex43wfqs-35sv9o

Plot HeunGPrime as a function of its third parameter :

https://wolfram.com/xid/0bni4ex43wfqs-vhxvag

Plot HeunGPrime as a function of and
:

https://wolfram.com/xid/0bni4ex43wfqs-f73ybn

https://wolfram.com/xid/0bni4ex43wfqs-8282mz

Plot the family of HeunGPrime functions for different accessory parameter :

https://wolfram.com/xid/0bni4ex43wfqs-dzixpt

https://wolfram.com/xid/0bni4ex43wfqs-dnzkk3

Differentiation (1)
The derivatives of HeunGPrime are calculated using the HeunG function:

https://wolfram.com/xid/0bni4ex43wfqs-6eb2k6

Integration (3)
Integral of HeunGPrime gives back HeunG:

https://wolfram.com/xid/0bni4ex43wfqs-ecaem6

Definite numerical integral of HeunGPrime:

https://wolfram.com/xid/0bni4ex43wfqs-3rkya0

More integrals with HeunGPrime:

https://wolfram.com/xid/0bni4ex43wfqs-gjk5w4


https://wolfram.com/xid/0bni4ex43wfqs-q3siwd

Series Expansions (4)
Taylor expansion for HeunGPrime at regular singular origin:

https://wolfram.com/xid/0bni4ex43wfqs-dux5ad

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

https://wolfram.com/xid/0bni4ex43wfqs-9rxgh1

Plots of the first three approximations for HeunGPrime around :

https://wolfram.com/xid/0bni4ex43wfqs-egrmm3

https://wolfram.com/xid/0bni4ex43wfqs-t525r

https://wolfram.com/xid/0bni4ex43wfqs-hrtnwe

Series expansion for HeunGPrime at any ordinary complex point:

https://wolfram.com/xid/0bni4ex43wfqs-ukhgue

Applications (1)Sample problems that can be solved with this function
Use the HeunGPrime function to calculate the derivatives of HeunG:

https://wolfram.com/xid/0bni4ex43wfqs-8yj5vx

Properties & Relations (3)Properties of the function, and connections to other functions
HeunGPrime is analytic at the origin:

https://wolfram.com/xid/0bni4ex43wfqs-mfv313

is a singular point of the HeunGPrime function:

https://wolfram.com/xid/0bni4ex43wfqs-epix0k

is a singular point of the HeunGPrime function:

https://wolfram.com/xid/0bni4ex43wfqs-5433ek

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

https://wolfram.com/xid/0bni4ex43wfqs-txs34a

HeunGPrime is the derivative of HeunG:

https://wolfram.com/xid/0bni4ex43wfqs-eqbaum

Possible Issues (2)Common pitfalls and unexpected behavior
HeunGPrime cannot be evaluated if is a nonpositive integer (so-called logarithmic cases):

https://wolfram.com/xid/0bni4ex43wfqs-8vqyfj


https://wolfram.com/xid/0bni4ex43wfqs-uuaubk


https://wolfram.com/xid/0bni4ex43wfqs-ru2szc

HeunGPrime is undefined when :

https://wolfram.com/xid/0bni4ex43wfqs-rwqzdr

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.
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.
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
Wolfram Language. (2020). HeunGPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeunGPrime.html
BibTeX
@misc{reference.wolfram_2025_heungprime, author="Wolfram Research", title="{HeunGPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunGPrime.html}", note=[Accessed: 11-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_heungprime, organization={Wolfram Research}, title={HeunGPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunGPrime.html}, note=[Accessed: 11-July-2025
]}