WeberE

WeberE[ν,z]

gives the Weber function .

WeberE[ν,μ,z]

gives the associated Weber function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • satisfies the differential equation .
  • is defined by .
  • WeberE[ν,z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, WeberE automatically evaluates to exact values.
  • WeberE can be evaluated to arbitrary numerical precision.
  • WeberE automatically threads over lists.

Examples

open allclose all

Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (31)

Numerical Evaluation  (4)

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:

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, {x, +, {i,  , y}}}, WeberE2]:

Plot the imaginary part of TemplateBox[{0, {x, +, {i,  , y}}}, WeberE2]:

Plot the real part of TemplateBox[{{-, {1, /, 4}}, {x, +, {i,  , y}}}, WeberE2]:

Plot the imaginary part of TemplateBox[{{-, {1, /, 4}}, {x, +, {i,  , y}}}, 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  (1)

Use FunctionExpand to expand WeberE into hypergeometric 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.

BibTeX

@misc{reference.wolfram_2021_webere, author="Wolfram Research", title="{WeberE}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/WeberE.html}", note=[Accessed: 18-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_webere, organization={Wolfram Research}, title={WeberE}, year={2008}, url={https://reference.wolfram.com/language/ref/WeberE.html}, note=[Accessed: 18-September-2021 ]}

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