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HeunGPrime
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HeunGPrime

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-3kfjq
Out[1]=1

Plot the HeunGPrime function:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-ftt82q
Out[1]=1

Series expansion of HeunGPrime:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-z8evs7
Out[1]=1

Scope  (28)Survey of the scope of standard use cases

Numerical Evaluation  (10)

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-uqurky
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-lw9h0n
Out[1]=1

HeunGPrime can take one or more complex number parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-64g5bd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-ft5oo1
Out[2]=2

HeunGPrime can take complex number arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-hunut5
Out[1]=1

Finally, HeunGPrime can take all complex number input:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-56m4mo
Out[1]=1

Evaluate HeunGPrime efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-2c7v5i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-yaawua
Out[2]=2

Lists and matrices:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-22a9kq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-1knfqv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bni4ex43wfqs-7yjugj
Out[3]=3

Evaluate HeunGPrime for points at branch cut to :

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-vush9d
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-zso2z1
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-thgd2
Out[1]=1

Or compute the matrix HeunGPrime function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-o5jpo
Out[2]=2

Specific Values  (5)

Value of HeunGPrime at origin:

In[1]:=1
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https://wolfram.com/xid/0bni4ex43wfqs-nuboa
Out[1]=1

Value of HeunGPrime at regular singular point is indeterminate:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-124w4g
Out[1]=1

Value of HeunGPrime at regular singular point is indeterminate:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-6iqn02
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-kqbmlj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-jzxssp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bni4ex43wfqs-i7w0vo
Out[3]=3

Value of HeunGPrime is not determined if :

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-n91pyz
Out[1]=1

Visualization  (5)

Plot the HeunGPrime function:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-n742f
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0bni4ex43wfqs-35sv9o
Out[1]=1

Plot HeunGPrime as a function of its third parameter :

In[1]:=1
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https://wolfram.com/xid/0bni4ex43wfqs-vhxvag
Out[1]=1

Plot HeunGPrime as a function of and :

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-f73ybn
In[2]:=2
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https://wolfram.com/xid/0bni4ex43wfqs-8282mz
Out[2]=2

Plot the family of HeunGPrime functions for different accessory parameter :

In[1]:=1
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https://wolfram.com/xid/0bni4ex43wfqs-dzixpt
In[2]:=2
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https://wolfram.com/xid/0bni4ex43wfqs-dnzkk3
Out[2]=2

Differentiation  (1)

The derivatives of HeunGPrime are calculated using the HeunG function:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-6eb2k6
Out[1]=1

Integration  (3)

Integral of HeunGPrime gives back HeunG:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-ecaem6
Out[1]=1

Definite numerical integral of HeunGPrime:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-3rkya0
Out[1]=1

More integrals with HeunGPrime:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-gjk5w4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-q3siwd
Out[2]=2

Series Expansions  (4)

Taylor expansion for HeunGPrime at regular singular origin:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-dux5ad
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-9rxgh1
Out[1]=1

Plots of the first three approximations for HeunGPrime around :

In[1]:=1
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https://wolfram.com/xid/0bni4ex43wfqs-egrmm3
In[2]:=2
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https://wolfram.com/xid/0bni4ex43wfqs-t525r
In[3]:=3
✖
https://wolfram.com/xid/0bni4ex43wfqs-hrtnwe
Out[3]=3

Series expansion for HeunGPrime at any ordinary complex point:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-ukhgue
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Use the HeunGPrime function to calculate the derivatives of HeunG:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-8yj5vx
Out[1]=1

Properties & Relations  (3)Properties of the function, and connections to other functions

HeunGPrime is analytic at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-mfv313
Out[1]=1

is a singular point of the HeunGPrime function:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-epix0k
Out[1]=1

is a singular point of the HeunGPrime function:

In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-5433ek
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bni4ex43wfqs-txs34a
Out[3]=3

HeunGPrime is the derivative of HeunG:

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-eqbaum
Out[1]=1

Possible Issues  (2)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-8vqyfj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bni4ex43wfqs-uuaubk
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bni4ex43wfqs-ru2szc
Out[3]=3

HeunGPrime is undefined when :

In[1]:=1
✖
https://wolfram.com/xid/0bni4ex43wfqs-rwqzdr
Out[1]=1
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.

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 ]}

✖
@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 ]}

✖
@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 ]}