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HeunDPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunD 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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HeunDPrime

HeunDPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunD function.

Details

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

Examples

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Basic Examples  (3)

Evaluate numerically:

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

Plot HeunDPrime:

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

Series expansion of HeunDPrime:

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

Scope  (24)

Numerical Evaluation  (9)

Evaluate to high precision:

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

HeunDPrime can take one or more complex number parameters:

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

HeunDPrime can take complex number arguments:

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

Finally, HeunDPrime can take all complex number input:

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

Evaluate HeunDPrime efficiently at high precision:

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

Lists and matrices:

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

Evaluate HeunDPrime for points on the real negative axis, bypassing irregular singular origin:

Wolfram Language code: HeunDPrime[-0.2 + I, -0.3, 0.12, 0.12, -0.32, -1.1]

Compute the elementwise values of an array:

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

Or compute the matrix HeunDPrime function using MatrixFunction:

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

Specific Values  (2)

Value of HeunDPrime at :

Wolfram Language code: HeunDPrime[q, α, γ, δ, ϵ, 1]

Value of HeunDPrime at origin is undetermined:

Wolfram Language code: HeunDPrime[12 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, 0]

Visualization  (5)

Plot the HeunDPrime function:

Wolfram Language code: Plot[HeunDPrime[4, -0.6, -0.7 , -0.18, 0.3 , z], {z, 3 / 10, 19 / 10}]

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

Wolfram Language code: Plot[Abs[HeunDPrime[4 + I, -0.6 + 0.9 I, -0.7 I, -0.18 - 0.03 I, 0.3  + 0.6 I, z]], {z, 3 / 10, 19 / 10}]

Plot HeunDPrime as a function of its second parameter :

Wolfram Language code: Plot[HeunDPrime[4, α, -0.7, -0.18, 0.3 , z] /. α -> {-2, Sqrt[20], 1 / 10}//Evaluate, {z, 3 / 10, 19 / 10}]

Plot HeunDPrime as a function of and :

Wolfram Language code: {α, γ, ϵ, δ} = {0.2 + I, -0.6 + 0.9 I, -0.7 I, 0.3  + 0.6 I};
Wolfram Language code: Plot3D[Abs[HeunDPrime[q, α, γ, δ, ϵ, z]], {q, -20, 2}, {z, 1 / 2, 2}, ColorFunction -> Function[{q, z, HDPrime}, Hue[HDPrime]], PlotRange -> All]

Plot the family of HeunDPrime functions for different accessory parameter :

Wolfram Language code: {α, γ, δ, ϵ} = {0.8 + 0.7 I, 0.9  - 0.6I, 0.2 - 0.7 I, -0.7 - 0.8I};
Wolfram Language code: Plot[Evaluate[Table[Abs[HeunDPrime[q, α, γ, δ, ϵ, z]], {q, -10, 4, 1}]], {z, 1 / 10, 19 / 10}, PlotStyle -> Table[{Hue[i / 20], Thickness[0.002]}, {i, 20}], PlotRange -> {0, 3}, Frame -> True, Axes -> False]

Differentiation  (1)

The derivatives of HeunDPrime are calculated using the HeunD function:

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

Integration  (3)

Integral of HeunDPrime gives back HeunD:

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

Definite numerical integral of HeunDPrime:

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

More integrals with HeunDPrime:

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

Series Expansions  (4)

Taylor expansion for HeunDPrime at point :

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

Coefficient of the second term in the series expansion of HeunDPrime at :

Wolfram Language code: SeriesCoefficient[HeunDPrime[q, α, γ, δ, ϵ, z], {z, 1, 2}]

Plots of the first three approximations for HeunDPrime 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[HeunDPrime[q, α, γ, δ, ϵ, z], {z, 1, m}], {m, 1, 3}];
Wolfram Language code: Plot[{HeunDPrime[q, α, γ, δ, ϵ, z], terms}, {z, 1 / 2, 2}, PlotRange -> All, PlotLegends -> {"HeunDPrime[q, α, γ, δ, ϵ, z]", "1st approximation", "2nd approximation", "3rd approximation"}]

Series expansion for HeunDPrime at any ordinary complex point:

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

Applications  (1)

Use the HeunDPrime function to calculate the derivatives of HeunD:

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

Properties & Relations  (3)

HeunDPrime is analytic at the point :

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

Origin is a singular point of the HeunDPrime function:

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

Except for this singular point, HeunDPrime can be calculated at any finite complex :

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

HeunDPrime is the derivative of HeunD:

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

Possible Issues  (1)

HeunDPrime diverges for big arguments:

Wolfram Language code: HeunDPrime[4.3 + I, 1.3, 1, 0.12, 4.32, 1000]
Wolfram Research (2020), HeunDPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunDPrime.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_heundprime, author="Wolfram Research", title="{HeunDPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunDPrime.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_heundprime, organization={Wolfram Research}, title={HeunDPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunDPrime.html}, note=[Accessed: 13-June-2026]}