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:

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

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (32)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

WeberE can be used with Interval and CenteredInterval objects:

Specific Values  (6)

Limiting value at infinity:

Values at zero:

WeberE for symbolic ν and x:

Find the first positive maximum of WeberE:

WeberE defined as StruveH for integer orders:

Evaluate WeberE for half-integer orders:

Visualization  (3)

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

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

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

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

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

Function Properties  (12)

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

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

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

The complex domain is the whole plane:

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

Use FullSimplify to simplify Weber functions:

WeberE threads elementwise over lists:

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

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

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

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

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

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

TraditionalForm formatting:

Differentiation  (3)

The first derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when ν=1/4:

The formula for the derivative with respect to z when ν=2:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The general term in the series expansion using SeriesCoefficient:

The Taylor expansion at a generic point:

Properties & Relations  (2)

Use FunctionExpand to expand WeberE into hypergeometric functions:

Relationships between the Anger and Weber functions:

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_2023_webere, author="Wolfram Research", title="{WeberE}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/WeberE.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_webere, organization={Wolfram Research}, title={WeberE}, year={2008}, url={https://reference.wolfram.com/language/ref/WeberE.html}, note=[Accessed: 19-March-2024 ]}