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WeberE[ν,z]

gives the Weber function TemplateBox[{nu, z}, WeberE2].

WeberE[ν,μ,z]

gives the associated Weber function TemplateBox[{nu, mu, z}, WeberE].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Differentiation and Integration  
Series Expansions  
Properties & Relations  
See Also
Related Guides
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WeberE

WeberE[ν,z]

gives the Weber function TemplateBox[{nu, z}, WeberE2].

WeberE[ν,μ,z]

gives the associated Weber function TemplateBox[{nu, mu, z}, WeberE].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{nu, z}, WeberE2] satisfies the differential equation .
  • TemplateBox[{nu, z}, WeberE2] is defined by TemplateBox[{nu, z}, WeberE2]=1/piint_0^pisin(theta nu-z sin(theta))dtheta.
  • WeberE[ν,z] is an entire function of z with no branch cut discontinuities.
  • TemplateBox[{nu, mu, z}, WeberE] is defined by TemplateBox[{nu, mu, z}, WeberE]=1/piint_0^pi(2sin(theta))^musin(theta nu-z sin(theta))dtheta.
  • For certain special arguments, WeberE automatically evaluates to exact values.
  • WeberE can be evaluated to arbitrary numerical precision.
  • WeberE automatically threads over lists.
  • WeberE can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Wolfram Language code: WeberE[1 / 2, 5.0]

Plot TemplateBox[{{1, /, 3}, x}, WeberE2] over a subset of the reals:

Wolfram Language code: ReImPlot[WeberE[1 / 3, x], {x, -25, 25}, PlotRange -> All]

Plot over a subset of the complexes:

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

Series expansion at the origin:

Wolfram Language code: Series[WeberE[1 / 2, x], {x, 0, 4}]//FullSimplify

Scope  (35)

Numerical Evaluation  (6)

Evaluate numerically:

Wolfram Language code: WeberE[1, 2.0]
Wolfram Language code: WeberE[0.5, -4.0]

Evaluate to high precision:

Wolfram Language code: N[WeberE[1, 3], 50]

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

Wolfram Language code: WeberE[2 / 3, 1.000000000240000000000000000000]

Complex number inputs:

Wolfram Language code: N[WeberE[1, 2 + I]]

Evaluate efficiently at high precision:

Wolfram Language code: WeberE[2, 4.`100]//Timing
Wolfram Language code: WeberE[2, 3`100];//Timing

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

Wolfram Language code: WeberE[0.3, 0.4, Interval[{0.5, 0.6}]]
Wolfram Language code: WeberE[2I, 1, CenteredInterval[2, 1 / 100]]

Or compute average-case statistical intervals using Around:

Wolfram Language code: WeberE[2, Around[2, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: WeberE[ 2, {{5 / 6, 0}, {1 / 2, -1 / 3}}]

Or compute the matrix WeberE function using MatrixFunction:

Wolfram Language code: MatrixFunction[WeberE[2, #]&, {{5 / 6, 0}, {1 / 2, -1 / 3}}]

Specific Values  (6)

Limiting value at infinity:

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

Values at zero:

Wolfram Language code: WeberE[1, 0]
Wolfram Language code: WeberE[0, 0]

WeberE for symbolic and :

Wolfram Language code: WeberE[ν, x]//FunctionExpand

Find the first positive maximum of WeberE:

Wolfram Language code: xmax = x /. FindRoot[D[WeberE[1 / 3, x ], x] == 0, {x, 6}]
Wolfram Language code: Plot[WeberE[1 / 3, x], {x, 0, 8}, Epilog -> Style[Point[{xmax, WeberE[1 / 3, xmax]}], PointSize[Large], Red]]

WeberE defined as StruveH for integer orders:

Wolfram Language code: WeberE[2, x]//FunctionExpand

Evaluate WeberE for half-integer orders:

Wolfram Language code: Table[WeberE[ν, x], {ν, {-1 / 3, 1 / 2}}]//FunctionExpand

Visualization  (3)

Plot the WeberE function for integer () and half-integer () orders:

Wolfram Language code: ReImPlot[{WeberE[0, x], WeberE[1, x], WeberE[-1 / 2, x]}, {x, -10, 10}]

Plot the real part of TemplateBox[{0, z}, WeberE2]:

Wolfram Language code: ComplexContourPlot[Re[WeberE[0, z]], {z, -4 - 4I, 8 + 4I}, Contours -> 24]

Plot the imaginary part of TemplateBox[{0, z}, WeberE2]:

Wolfram Language code: ComplexContourPlot[Im[WeberE[0, z]], {z, -4 - 4I, 8 + 4I}, Contours -> 24]

Plot the real part of TemplateBox[{{-, {1, /, 4}}, z}, WeberE2]:

Wolfram Language code: ComplexContourPlot[Re[WeberE[-1 / 4, z]], {z, -4 - 4I, 8 + 4I}, IconizedObject[«PlotOptions»]]

Plot the imaginary part of TemplateBox[{{-, {1, /, 4}}, z}, WeberE2]:

Wolfram Language code: ComplexContourPlot[Im[WeberE[-1 / 4, z]], {z, -4 - 4I, 8 + 4I}, IconizedObject[«PlotOptions»]]

Function Properties  (12)

The real domain of TemplateBox[{0, x}, WeberE2]:

Wolfram Language code: FunctionDomain[WeberE[0, x], x]

The complex domain of TemplateBox[{0, z}, WeberE2]:

Wolfram Language code: FunctionDomain[WeberE[0, z], z, Complexes]

TemplateBox[{{1, /, 2}, x}, WeberE2] is defined for all real values:

Wolfram Language code: FunctionDomain[WeberE[1 / 2, x], x]

The complex domain is the whole plane:

Wolfram Language code: FunctionDomain[WeberE[1 / 2, z], z, Complexes]

The approximate function range of TemplateBox[{1, z}, WeberE2]:

Wolfram Language code: FunctionRange[WeberE[1, x], x, y]//Quiet

Use FullSimplify to simplify Weber functions:

Wolfram Language code: FullSimplify[x WeberE[2, x] + x WeberE[0, x]]

WeberE threads elementwise over lists:

Wolfram Language code: WeberE[0.3, {0.1, 0.2, 0.3}]

TemplateBox[{{1, /, 2}, x}, WeberE2] is an analytic function of :

Wolfram Language code: FunctionAnalytic[WeberE[1 / 2, x], x]

TemplateBox[{{1, /, 2}, x}, WeberE2] is neither non-decreasing nor non-increasing:

Wolfram Language code: FunctionMonotonicity[WeberE[1 / 2, x], x]

TemplateBox[{{1, /, 2}, x}, WeberE2] is not injective:

Wolfram Language code: FunctionInjective[WeberE[1 / 2, x], x]
Wolfram Language code: Plot[{WeberE[1 / 2, x], .1}, {x, -10, 10}]

TemplateBox[{{1, /, 2}, x}, WeberE2] is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[WeberE[1 / 2, x], x]

TemplateBox[{{1, /, 2}, x}, WeberE2] has neither singularities nor discontinuities:

Wolfram Language code: FunctionSingularities[WeberE[1 / 2, x], x]
Wolfram Language code: FunctionDiscontinuities[WeberE[1 / 2, x], x]

TemplateBox[{{1, /, 2}, x}, WeberE2] is neither convex nor concave:

Wolfram Language code: FunctionConvexity[WeberE[1 / 2, x], x]

TraditionalForm formatting:

Wolfram Language code: WeberE[ν, x]//TraditionalForm
Wolfram Language code: WeberE[ν, μ, x]//TraditionalForm

Differentiation and Integration  (5)

The first derivatives with respect to :

Wolfram Language code: D[WeberE[ν, x], x]
Wolfram Language code: D[WeberE[ν, μ, x], x]

Higher derivatives with respect to :

Wolfram Language code: Table[D[WeberE[ν, x], {x, k}], {k, 1, 3}]//FullSimplify

Plot the higher derivatives with respect to when :

Wolfram Language code: Plot[Evaluate[% /. {ν -> 1 / 4}], {x, -10, 10}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]

The formula for the derivative with respect to when :

Wolfram Language code: D[WeberE[2, x], {x, k}]// FullSimplify

Indefinite integral of WeberE:

Wolfram Language code: Integrate[WeberE[ν, x], x]//FullSimplify
Wolfram Language code: Integrate[WeberE[ν, μ, x], x]//FullSimplify

More integrals:

Wolfram Language code: ∫ x WeberE[ν, x]ⅆx//TraditionalForm
Wolfram Language code: ∫ x^αWeberE[ν, μ, x^2]ⅆx//TraditionalForm

Series Expansions  (3)

Find the Taylor expansion using Series:

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

Plots of the first three approximations around :

Wolfram Language code: terms = Normal@Table[Series[WeberE[1 / 3, x], {x, 0, m}], {m, 1, 5, 2}]; Plot[{WeberE[1 / 3, x], terms}, {x, -10, 10}, PlotRange -> {-10, 10}]

The general term in the series expansion using SeriesCoefficient:

Wolfram Language code: SeriesCoefficient[WeberE[ν, x], {x, 1, n}]

The Taylor expansion at a generic point:

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

Properties & Relations  (2)

Use FunctionExpand to expand WeberE into hypergeometric functions:

Wolfram Language code: FunctionExpand[WeberE[ν, x]]

Relationships between the Anger and Weber functions:

Wolfram Language code: FunctionExpand[Sin[Pi ν]AngerJ[ν, x] == Cos[Pi ν]WeberE[ν, x] - WeberE[-ν, x]]//Simplify
Wolfram Language code: FunctionExpand[Sin[Pi ν]WeberE[ν, x] == -Cos[Pi ν]AngerJ[ν, x] + AngerJ[-ν, x]]//Simplify
Wolfram Research (2008), WeberE, Wolfram Language function, https://reference.wolfram.com/language/ref/WeberE.html.

Text

Wolfram Research (2008), WeberE, Wolfram Language function, https://reference.wolfram.com/language/ref/WeberE.html.

CMS

Wolfram Language. 2008. "WeberE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeberE.html.

APA

Wolfram Language. (2008). WeberE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeberE.html

BibTeX

@misc{reference.wolfram_2026_webere, author="Wolfram Research", title="{WeberE}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/WeberE.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_webere, organization={Wolfram Research}, title={WeberE}, year={2008}, url={https://reference.wolfram.com/language/ref/WeberE.html}, note=[Accessed: 13-June-2026]}