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HeunBPrime
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HeunBPrime

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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)Summary of the most common use cases

Evaluate numerically:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-3kfjq
Direct link to example
Out[1]=1

Plot HeunBPrime:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-ftt82q
Direct link to example
Out[1]=1

Series expansion of HeunBPrime:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-z8evs7
Direct link to example
Out[1]=1

Scope  (22)Survey of the scope of standard use cases

Numerical Evaluation  (8)

Evaluate to high precision:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-uqurky
Direct link to example
Out[1]=1

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

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-lw9h0n
Direct link to example
Out[1]=1

HeunBPrime can take one or more complex number parameters:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-64g5bd
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-ft5oo1
Direct link to example
Out[2]=2

HeunBPrime can take complex number arguments:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-hunut5
Direct link to example
Out[1]=1

Finally, HeunBPrime can take all complex number input:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-56m4mo
Direct link to example
Out[1]=1

Evaluate HeunBPrime efficiently at high precision:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-2c7v5i
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-yaawua
Direct link to example
Out[2]=2

Lists and matrices:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-22a9kq
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-1knfqv
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0mlhtf8zwp0k-7yjugj
Direct link to example
Out[3]=3

Compute the elementwise values of an array:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-thgd2
Direct link to example
Out[1]=1

Or compute the matrix HeunBPrime function using MatrixFunction:

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In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-o5jpo
Direct link to example
Out[2]=2

Specific Values  (1)

Value of HeunBPrime at origin:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-nuboa
Direct link to example
Out[1]=1

Visualization  (5)

Plot the HeunBPrime function:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-n742f
Direct link to example
Out[1]=1

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

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-35sv9o
Direct link to example
Out[1]=1

Plot HeunBPrime as a function of its second parameter :

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-vhxvag
Direct link to example
Out[1]=1

Plot HeunBPrime as a function of and :

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-wsjihm
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-8282mz
Direct link to example
Out[2]=2

Plot the family of HeunBPrime functions for different accessory parameter :

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-kjmc5h
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-dnzkk3
Direct link to example
Out[2]=2

Differentiation  (1)

The derivatives of HeunBPrime are calculated using the HeunB function:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-6eb2k6
Direct link to example
Out[1]=1

Integration  (3)

Integral of HeunBPrime gives back HeunB:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-ecaem6
Direct link to example
Out[1]=1

Definite numerical integral of HeunBPrime:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-3rkya0
Direct link to example
Out[1]=1

More integrals with HeunBPrime:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-gjk5w4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-q3siwd
Direct link to example
Out[2]=2

Series Expansions  (4)

Taylor expansion for HeunBPrime at regular singular origin:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-dux5ad
Direct link to example
Out[1]=1

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

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-9rxgh1
Direct link to example
Out[1]=1

Plots of the first three approximations for HeunBPrime around :

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-n2oba8
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0mlhtf8zwp0k-4lctoi
Direct link to example
Copy to clipboard.
In[3]:=3
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https://wolfram.com/xid/0mlhtf8zwp0k-hrtnwe
Direct link to example
Out[3]=3

Series expansion for HeunBPrime at any ordinary complex point:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-ukhgue
Direct link to example
Out[1]=1

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

Use the HeunBPrime function to calculate the derivatives of HeunB:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-8yj5vx
Direct link to example
Out[1]=1

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

HeunBPrime is analytic at the origin:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-usyc66
Direct link to example
Out[1]=1

HeunBPrime can be calculated at any finite complex :

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-txs34a
Direct link to example
Out[1]=1

HeunBPrime is the derivative of HeunB:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-rvrd6q
Direct link to example
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

HeunBPrime diverges for big arguments:

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In[1]:=1
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https://wolfram.com/xid/0mlhtf8zwp0k-6fjqjs
Direct link to example
Out[1]=1
Wolfram Research (2020), HeunBPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunBPrime.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_heunbprime, author="Wolfram Research", title="{HeunBPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunBPrime.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_heunbprime, author="Wolfram Research", title="{HeunBPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunBPrime.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_heunbprime, organization={Wolfram Research}, title={HeunBPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunBPrime.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_heunbprime, organization={Wolfram Research}, title={HeunBPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunBPrime.html}, note=[Accessed: 26-March-2025 ]}