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ScorerHi[z]

gives the Scorer function TemplateBox[{z}, ScorerHi].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
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Function Properties  
Differentiation and Integration  
Series Expansions  
Function Identities and Simplifications  
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ScorerHi

ScorerHi[z]

gives the Scorer function TemplateBox[{z}, ScorerHi].

Details

  • ScorerHi is also known as an inhomogeneous Airy function.
  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • The Scorer function TemplateBox[{z}, ScorerHi] is a solution to the inhomogeneous Airy differential equation .
  • TemplateBox[{z}, ScorerHi] tends to zero as .
  • ScorerHi[z] is an entire function of z with no branch cut discontinuities.
  • For certain arguments, ScorerHi automatically evaluates to exact values.
  • ScorerHi can be evaluated to arbitrary numerical precision.
  • ScorerHi automatically threads over lists.
  • ScorerHi can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Wolfram Language code: ScorerHi[3.]

Plot over a subset of the reals:

Wolfram Language code: Plot[ScorerHi[x], {x, -10, 3}]

Plot over a subset of the complexes:

Wolfram Language code: ComplexPlot3D[ScorerHi[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]

Series expansion at the origin:

Wolfram Language code: Series[ScorerHi[x], {x, 0, 3}]

Scope  (32)

Numerical Evaluation  (6)

Evaluate numerically:

Wolfram Language code: ScorerHi[7.]
Wolfram Language code: N[ScorerHi[Pi]]

Evaluate to high precision:

Wolfram Language code: N[ScorerHi[1 / 3], 50]

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

Wolfram Language code: ScorerHi[0.312785550000555523000]

Complex number inputs:

Wolfram Language code: ScorerHi[0.2 + I]
Wolfram Language code: N[ScorerHi[I Pi / 3]]

Evaluate efficiently at high precision:

Wolfram Language code: N[ScorerHi[Pi / 3`100]]//Timing
Wolfram Language code: N[ScorerHi[Pi / 4`10000]];//Timing

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Wolfram Language code: ScorerHi[Interval[{0.5, 0.6}]]
Wolfram Language code: ScorerHi[CenteredInterval[2, 1 / 100]]

Or compute average-case statistical intervals using Around:

Wolfram Language code: ScorerHi[Around[.9, 0.1]]

Compute the elementwise values of an array:

Wolfram Language code: ScorerHi[{{1.3, .5}, {0, 0.2}}]

Or compute the matrix ScorerHi function using MatrixFunction:

Wolfram Language code: MatrixFunction[ScorerHi[#]&, {{1.3, .5}, {0, 0.2}}]

Specific Values  (3)

Simple exact values are generated automatically:

Wolfram Language code: ScorerHi[0]

Limiting values at infinity:

Wolfram Language code: Limit[ScorerHi[x], x -> Infinity]

Find a value of x for which ScorerHi[x]=4:

Wolfram Language code: xval = x /. FindRoot[ScorerHi[x] == 4, {x, 2}]
Wolfram Language code: Plot[ScorerHi[x ], {x, 0, 3}, Epilog -> Style[Point[{xval, ScorerHi[xval ]}], PointSize[Large], Red]]

Visualization  (2)

Plot the ScorerHi function:

Wolfram Language code: Plot[ScorerHi[x], {x, -10, 2}]

Plot the real part of TemplateBox[{z}, ScorerHi]:

Wolfram Language code: ComplexContourPlot[Re[ScorerHi[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 24]

Plot the imaginary part of TemplateBox[{z}, ScorerHi]:

Wolfram Language code: ComplexContourPlot[Im[ScorerHi[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 24]

Function Properties  (11)

Real domain of ScorerHi:

Wolfram Language code: FunctionDomain[ScorerHi[x], x]

Complex domain:

Wolfram Language code: FunctionDomain[ScorerHi[z], z, Complexes]

Approximate function range of ScorerHi:

Wolfram Language code: FunctionRange[ScorerHi[x], x, y]//Quiet

ScorerHi threads elementwise over lists:

Wolfram Language code: ScorerHi[{-1., 1., 2.}]

ScorerHi is an analytic function of x:

Wolfram Language code: FunctionAnalytic[ScorerHi[x], x]

ScorerHi is nondecreasing:

Wolfram Language code: FunctionMonotonicity[ScorerHi[x], x]

ScorerHi is injective:

Wolfram Language code: FunctionInjective[ScorerHi[x], x]
Wolfram Language code: Plot[{ScorerHi[x], 1}, {x, -10, 5}]

ScorerHi is not surjective:

Wolfram Language code: FunctionSurjective[ScorerHi[x], x]
Wolfram Language code: Plot[{ScorerHi[x], -1}, {x, -10, 5}]

ScorerHi is non-negative:

Wolfram Language code: FunctionSign[ScorerHi[x], x]

ScorerHi does not have either singularity or discontinuity:

Wolfram Language code: FunctionSingularities[ScorerHi[x], x]
Wolfram Language code: FunctionDiscontinuities[ScorerHi[x], x]

ScorerHi is convex:

Wolfram Language code: FunctionConvexity[ScorerHi[x], x]

TraditionalForm typesetting:

Wolfram Language code: ScorerHi[z]//TraditionalForm

Differentiation and Integration  (5)

First derivative with respect to z:

Wolfram Language code: D[ScorerHi[z] , z]

Higher derivatives with respect to z:

Wolfram Language code: Table[D[ScorerHi[z], {z, k}], {k, 1, 3}]//Simplify

Plot the higher derivatives with respect to z:

Wolfram Language code: Plot[%, {z, -3, 5}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]

Formula for the ^(th) derivative with respect to z:

Wolfram Language code: D[ScorerHi[z], {z, k}]// FullSimplify

Indefinite integral of ScorerHi:

Wolfram Language code: Integrate[ScorerHi[x], x]//FullSimplify

More integrals:

Wolfram Language code: ∫ z^αScorerHi[z^2]ⅆz//TraditionalForm
Wolfram Language code: ∫ ScorerHi[z]ScorerHiPrime[z]ⅆz//TraditionalForm

Series Expansions  (2)

Find the Taylor expansion using Series:

Wolfram Language code: Series[ScorerHi[x], {x, 0, 3}]

Plots of the first three approximations around :

Wolfram Language code: terms = Normal@Table[Series[ScorerHi[x], {x, 0, m}], {m, 1, 5, 2}]; Plot[{ScorerHi[x], terms}, {x, -5, 5}, PlotRange -> {-5, 5}]

Taylor expansion at a generic point:

Wolfram Language code: Series[ScorerHi[x], {x, x0, 2}]// FullSimplify

Function Identities and Simplifications  (3)

FunctionExpand tries to simplify the argument of ScorerHi:

Wolfram Language code: FunctionExpand[ScorerHi[z E^2 π I]]//Simplify
Wolfram Language code: FunctionExpand[ScorerHi[(z^2)^1 / 2]]//Simplify

Functional identity:

Wolfram Language code: ScorerHi[z] + E^2 π I ScorerHi[z E^2 π I] + E^-2 π I ScorerHi[z E^-2 π I]//FullSimplify

ScorerHi can be represented as a DifferentialRoot:

Wolfram Language code: DifferentialRootReduce[ScorerHi[x], x]
Wolfram Research (2014), ScorerHi, Wolfram Language function, https://reference.wolfram.com/language/ref/ScorerHi.html.

Text

Wolfram Research (2014), ScorerHi, Wolfram Language function, https://reference.wolfram.com/language/ref/ScorerHi.html.

CMS

Wolfram Language. 2014. "ScorerHi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ScorerHi.html.

APA

Wolfram Language. (2014). ScorerHi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ScorerHi.html

BibTeX

@misc{reference.wolfram_2026_scorerhi, author="Wolfram Research", title="{ScorerHi}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ScorerHi.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_scorerhi, organization={Wolfram Research}, title={ScorerHi}, year={2014}, url={https://reference.wolfram.com/language/ref/ScorerHi.html}, note=[Accessed: 13-June-2026]}