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HeunBPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunB function.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Differentiation  
Integration  
Series Expansions  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
Related Links
History
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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:

Wolfram Language code: HeunBPrime[1.2, -0.6, -0.7 , -0.18, 0.3, 0.12]

Plot HeunBPrime:

Wolfram Language code: Plot[HeunBPrime[1.2, -0.6, -0.7 , -0.18, 0.3 , x], {x, -3, 1}]

Series expansion of HeunBPrime:

Wolfram Language code: Series[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 0, 1}]

Scope  (22)

Numerical Evaluation  (8)

Evaluate to high precision:

Wolfram Language code: N[HeunBPrime[9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, 1 / 10], 50]

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

Wolfram Language code: HeunBPrime[9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, 0.10000000000000000001]

HeunBPrime can take one or more complex number parameters:

Wolfram Language code: HeunBPrime[1.2 + I, -1.4, 0.12, 0.03, -0.2, 0.1]
Wolfram Language code: HeunBPrime[1.2 + I, -1.4 + 0.I, 0.12 - I, 0.03 + 0.123I, -0.2 - 0.2I, 0.1]

HeunBPrime can take complex number arguments:

Wolfram Language code: HeunBPrime[1.2, -1.4, 0.12, 0.03, -0.2, 0.1 + I]

Finally, HeunBPrime can take all complex number input:

Wolfram Language code: HeunBPrime[1.2 + I, -1.4 + 0.I, 0.12 - I, 0.03 + 0.123I, -0.2 - 0.2I, 0.1 + I]

Evaluate HeunBPrime efficiently at high precision:

Wolfram Language code: HeunBPrime[1 / 2, -1 / 3, 1 / 4, 1 / 5, -1 / 6, 1 / 7`100]//Timing
Wolfram Language code: HeunBPrime[1 / 2, -1 / 3, 1 / 4, 1 / 5, -1 / 6, 1 / 7 + I / 2`100]//Timing

Lists and matrices:

Wolfram Language code: HeunBPrime[1.2, -1.4, 0.12, 0.03, -0.2, {0.15, 0.1 + I, I, 4}]
Wolfram Language code: HeunBPrime[-0.002, {1.3, -0.4}, 0.12, -0.14, 4.32, -1.4]
Wolfram Language code: HeunBPrime[1.2, -1.4, 0.12, 0.03, -0.2, (| | | | :- | :---- | | π | u | | v | (π/2) |) ]

Compute the elementwise values of an array:

Wolfram Language code: HeunBPrime[.1 + I, I, 0.12, 0, -0.32, {{1 / 2, -1}, {1, 1 / 2}}]

Or compute the matrix HeunBPrime function using MatrixFunction:

Wolfram Language code: MatrixFunction[HeunBPrime[.1 + I, I, 0.12, 0, -0.32, #]&, {{1 / 2, -1}, {0, 1 / 2}}]

Specific Values  (1)

Value of HeunBPrime at origin:

Wolfram Language code: HeunBPrime[q, α, γ, δ, ϵ, 0]

Visualization  (5)

Plot the HeunBPrime function:

Wolfram Language code: Plot[HeunBPrime[4, -0.6, -0.7 , -0.18, 0.3 , z], {z, -3 / 10, 9 / 10}]

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

Wolfram Language code: Plot[Abs[HeunBPrime[4 + I, -0.6 + 0.9 I, -0.7 I, -0.18 - 0.03 I, 0.3  + 0.6 I, z]], {z, -3 / 10, 9 / 10}]

Plot HeunBPrime as a function of its second parameter :

Wolfram Language code: Plot[HeunBPrime[4, α, -0.7, -0.18, 0.3 , z] /. α -> {-2, Sqrt[20], 1 / 10}//Evaluate, {z, -3 / 10, 9 / 10}]

Plot HeunBPrime as a function of and :

Wolfram Language code: {q, γ, ϵ, δ} = {0.2 + I, -0.6 + 0.9 I, -0.7 I, 0.3  + 0.6 I};
Wolfram Language code: Plot3D[Abs[HeunBPrime[q, α, γ, δ, ϵ, z]], {α, -10, 2}, {z, 1 / 10, 9 / 10}, ColorFunction -> Function[{q, z, HBPrime}, Hue[HBPrime]], PlotRange -> All]

Plot the family of HeunBPrime functions for different accessory parameter :

Wolfram Language code: {α, γ, δ, ϵ} = {0.7  - 0.9 I, 0.3 - 0.5 I, -0.4 + 0.8 I, -0.3 - 0.6 I};
Wolfram Language code: Plot[Evaluate[Table[Abs[HeunBPrime[q, α, γ, δ, ϵ, z]], {q, -16, 12, 3}]], {z, -6, 99 / 100}, PlotStyle -> Table[{Hue[i / 10], Thickness[0.002]}, {i, 20}], PlotRange -> {0, 8}, Frame -> True, Axes -> False]

Differentiation  (1)

The derivatives of HeunBPrime are calculated using the HeunB function:

Wolfram Language code: D[HeunBPrime[q, α, γ, δ, ϵ, z], z]

Integration  (3)

Integral of HeunBPrime gives back HeunB:

Wolfram Language code: Integrate[HeunBPrime[q, α, γ, δ, ϵ, z], z]

Definite numerical integral of HeunBPrime:

Wolfram Language code: NIntegrate[HeunBPrime[12 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, 0, 1 / 3}]

More integrals with HeunBPrime:

Wolfram Language code: NIntegrate[z ^ 2 HeunBPrime[12 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, -1, 1 / 3}]
Wolfram Language code: NIntegrate[Sin[Sqrt[z]] ^ 2 HeunBPrime[12 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, -1, 1 / 3}]

Series Expansions  (4)

Taylor expansion for HeunBPrime at regular singular origin:

Wolfram Language code: Series[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 0, 1}]

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

Wolfram Language code: SeriesCoefficient[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 0, 1}]

Plots of the first three approximations for HeunBPrime around :

Wolfram Language code: {q, α, γ, δ, ϵ} = {1 / 31, 9 / 10, 1 / 10, 14 / 10, 1 / 2};
Wolfram Language code: terms = Normal@Table[Series[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 0, m}], {m, 1, 3}];
Wolfram Language code: Plot[{HeunBPrime[q, α, γ, δ, ϵ, z], terms}, {z, -1, 2}, PlotRange -> All, PlotLegends -> {"HeunBPrime[q, α, γ, δ, ϵ, z]", "1st approximation", "2nd approximation", "3rd approximation"}]

Series expansion for HeunBPrime at any ordinary complex point:

Wolfram Language code: Series[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 1 / 2, 1}]//FullSimplify

Applications  (1)

Use the HeunBPrime function to calculate the derivatives of HeunB:

Wolfram Language code: D[HeunB[q, α, γ, δ, ϵ, z], {z, 2}]

Properties & Relations  (3)

HeunBPrime is analytic at the origin:

Wolfram Language code: Series[HeunBPrime[q, α, γ, δ, ϵ, z], {z, 0, 1}]

HeunBPrime can be calculated at any finite complex :

Wolfram Language code: HeunBPrime[4 + I, -1 / 2, 1 / 4, -7 / 5, 2, z] /. z -> RandomComplex[{-2 - I, 2 + I}, 5]

HeunBPrime is the derivative of HeunB:

Wolfram Language code: D[HeunB[q, α, γ, δ, ϵ, z], z]

Possible Issues  (1)

HeunBPrime diverges for big arguments:

Wolfram Language code: HeunBPrime[0.03 + I, 1.3, 1, 0.12, 4.32, 150]
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_2026_heunbprime, author="Wolfram Research", title="{HeunBPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunBPrime.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_heunbprime, organization={Wolfram Research}, title={HeunBPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunBPrime.html}, note=[Accessed: 12-June-2026]}