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HeunGPrime[a,q,α,β,γ,δ,z]

gives the -derivative of the HeunG 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
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HeunGPrime

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

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Basic Examples  (3)

Evaluate numerically:

Wolfram Language code: HeunGPrime[4.3 + 0.1 I, -0.2, 1.3, 0.12, -0.14, 4.32, 0.12]

Plot the HeunGPrime function:

Wolfram Language code: Plot[HeunGPrime[4.3, -0.2, 1.3, 0.12, -0.14, 4.32, x], {x, -0.5, 0.5}]

Series expansion of HeunGPrime:

Wolfram Language code: Series[HeunGPrime[a, q, α, β, γ, δ, z], {z, 0, 1}]

Scope  (28)

Numerical Evaluation  (10)

Evaluate to high precision:

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

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

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

HeunGPrime can take one or more complex number parameters:

Wolfram Language code: HeunGPrime[1.2 + I, 0.9, -1.4, 0.12, 0.03, -0.2, 0.1]
Wolfram Language code: HeunGPrime[1.2 + I, 0.9 - 0.1I, -1.4 + 0.I, 0.12 - I, 0.03 + 0.123I, -0.2 - 0.2I, 0.1]

HeunGPrime can take complex number arguments:

Wolfram Language code: HeunGPrime[1.2, 0.9, -1.4, 0.12, 0.03, -0.2, 0.1 + I]

Finally, HeunGPrime can take all complex number input:

Wolfram Language code: HeunGPrime[1.2 + I, 0.9 - 0.1I, -1.4 + 0.I, 0.12 - I, 0.03 + 0.123I, -0.2 - 0.2I, 0.1 + I]

Evaluate HeunGPrime efficiently at high precision:

Wolfram Language code: HeunGPrime[1 / 2, 1 / 3, -1 / 4, 1 / 5, 1 / 6, -1 / 7, 1 / 8`100]//Timing
Wolfram Language code: HeunGPrime[1 / 2, 1 / 3, -1 / 4, 1 / 5, 1 / 6, -1 / 7, 1 / 8 + I / 2`100]//Timing

Lists and matrices:

Wolfram Language code: HeunGPrime[1.2, 0.9, -1.4, 0.12, 0.03, -0.2, {0.1, 0.1 + I, I, 4}]
Wolfram Language code: HeunGPrime[4.3, -0.2, {1.3, -0.4}, 0.12, -0.14, 4.32, -1.4]
Wolfram Language code: HeunGPrime[1.2, 0.9, -1.4, 0.12, 0.03, -0.2, (| | | | :- | :---- | | π | u | | v | (π/2) |) ]

Evaluate HeunGPrime for points at branch cut to :

Wolfram Language code: HeunGPrime[4.3 + I, 1.3, -0.71, 0.12, 1, 4.32, 11]

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

Wolfram Language code: With[{a = 4.3 + I}, HeunGPrime[a, 1.3, -0.71, 0.12, 1, 4.32, 2a]]

Compute the elementwise values of an array:

Wolfram Language code: HeunGPrime[12.1, 9.1, -14, 12, 1, -2, {{.2, 0}, {0, .2}}]

Or compute the matrix HeunGPrime function using MatrixFunction:

Wolfram Language code: MatrixFunction[HeunGPrime[12.1, 9.1, -14, 12, 1, -2, #]&, {{.2, 0}, {0, .2}}]

Specific Values  (5)

Value of HeunGPrime at origin:

Wolfram Language code: HeunGPrime[a, q, α, β, γ, δ, 0]

Value of HeunGPrime at regular singular point is indeterminate:

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

Value of HeunGPrime at regular singular point is indeterminate:

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

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

Wolfram Language code: HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, -3, -2 / 10, 2 / 10]
Wolfram Language code: HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, 0, -2 / 10, 2 / 10]
Wolfram Language code: HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, 1, -2 / 10, 2 / 10]//N

Value of HeunGPrime is not determined if :

Wolfram Language code: HeunGPrime[0, 9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, 2 / 10]

Visualization  (5)

Plot the HeunGPrime function:

Wolfram Language code: Plot[HeunGPrime[4, -20, -0.6, -0.7 , -0.18, 0.3 , z], {z, -3 / 10, 9 / 10}]

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

Wolfram Language code: Plot[Abs[HeunGPrime[4 + I, -20, -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 HeunGPrime as a function of its third parameter :

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

Plot HeunGPrime as a function of and :

Wolfram Language code: {a, α, β, γ, δ} = {4 + I, -0.6 + 0.9 I, -0.7 I, -0.18 - 0.03 I, 0.3  + 0.6 I};
Wolfram Language code: Plot3D[Abs[HeunGPrime[a, q, α, β, γ, δ, z]], {q, -20, -1}, {z, -1 / 10, 9 / 10}, ColorFunction -> Function[{q, z, HGPrime}, Hue[HGPrime]], PlotRange -> All]

Plot the family of HeunGPrime functions for different accessory parameter :

Wolfram Language code: {a, α, β, γ, δ} = {4 + I, -0.6 + 0.9 I, -0.7 I, -0.18 - 0.03 I, 0.3  + 0.6 I};
Wolfram Language code: Plot[Evaluate[Table[Abs[HeunGPrime[a, q, α, β, γ, δ, z]], {q, -20, -3, 1}]], {z, -1 / 2, 9 / 10}, PlotStyle -> Table[{Hue[i / 20], Thickness[0.002]}, {i, 20}], PlotRange -> All, Frame -> True, Axes -> False]

Differentiation  (1)

The derivatives of HeunGPrime are calculated using the HeunG function:

Wolfram Language code: D[HeunGPrime[a, q, α, β, γ, δ, z], z]

Integration  (3)

Integral of HeunGPrime gives back HeunG:

Wolfram Language code: Integrate[HeunGPrime[a, q, α, β, γ, δ, z], z]

Definite numerical integral of HeunGPrime:

Wolfram Language code: NIntegrate[HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, 0, 1 / 3}]

More integrals with HeunGPrime:

Wolfram Language code: NIntegrate[z ^ 2 HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, -1, 1 / 3}]
Wolfram Language code: NIntegrate[Sin[Sqrt[z]] ^ 2 HeunGPrime[12 / 10, 9 / 10, -14 / 10, 12 / 100, 3 / 100, -2 / 10, z], {z, -1, 1 / 3}]

Series Expansions  (4)

Taylor expansion for HeunGPrime at regular singular origin:

Wolfram Language code: Series[HeunGPrime[a, q, α, β, γ, δ, z], {z, 0, 1}]

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

Wolfram Language code: SeriesCoefficient[HeunGPrime[a, q, α, β, γ, δ, z], {z, 0, 1}]

Plots of the first three approximations for HeunGPrime around :

Wolfram Language code: {a, q, α, β, γ, δ} = {12 / 10, 9 / 10, 14 / 10, 12 / 100, 3 / 100, -2 / 10};
Wolfram Language code: terms = Normal@Table[Series[HeunGPrime[a, q, α, β, γ, δ, z], {z, 0, m}], {m, 1, 3}];
Wolfram Language code: Plot[{HeunGPrime[a, q, α, β, γ, δ, z], terms}//Evaluate, {z, -1, 1 / 2}, PlotRange -> All, PlotLegends -> {"HeunGPrime[a, q, α, β, γ, δ, z]", "1st approximation", "2nd approximation", "3rd approximation"}]

Series expansion for HeunGPrime at any ordinary complex point:

Wolfram Language code: Series[HeunGPrime[a, q, α, β, γ, δ, z], {z, 1 / 2, 1}]//FullSimplify

Applications  (1)

Use the HeunGPrime function to calculate the derivatives of HeunG:

Wolfram Language code: D[HeunG[a, q, α, β, γ, δ, z], {z, 2}]

Properties & Relations  (3)

HeunGPrime is analytic at the origin:

Wolfram Language code: Series[HeunGPrime[a, q, α, β, γ, δ, z], {z, 0, 1}]

is a singular point of the HeunGPrime function:

Wolfram Language code: HeunGPrime[a, q, α, β, γ, δ, 1]

is a singular point of the HeunGPrime function:

Wolfram Language code: HeunGPrime[a, q, α, β, γ, δ, a]

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

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

HeunGPrime is the derivative of HeunG:

Wolfram Language code: D[HeunG[a, q, α, β, γ, δ, z], z]

Possible Issues  (2)

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

Wolfram Language code: HeunGPrime[4.3 + I, -14.2, 1.3, 0.12, 1.1, 4.32, 4.3 + 0.5I]
Wolfram Language code: HeunGPrime[4.3 + I, -14.2, 1.3, 0.12, 0, 4.32, 4.3 + 0.5I]
Wolfram Language code: HeunGPrime[4.3 + I, -14.2, 1.3, 0.12, -400, 4.32, (4.3 + I) * 2]

HeunGPrime is undefined when :

Wolfram Language code: HeunGPrime[0, q, α, β, γ, δ, z]
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.

CMS

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

BibTeX

@misc{reference.wolfram_2026_heungprime, author="Wolfram Research", title="{HeunGPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunGPrime.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_heungprime, organization={Wolfram Research}, title={HeunGPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunGPrime.html}, note=[Accessed: 12-June-2026]}