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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 $TemplateBox[{nu, z}, WeberE2]=1/piint_0^pisin(theta nu-z sin(theta))dtheta$ .
WeberE[ν,z] is an entire function of z with no branch cut discontinuities.
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 over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope (35)

Numerical Evaluation (6)

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:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix WeberE function using MatrixFunction:

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 :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

Function Properties (12)

The real domain of :

The complex domain of :

is defined for all real values:

The complex domain is the whole plane:

The approximate function range of :

Use FullSimplify to simplify Weber functions:

WeberE threads elementwise over lists:

is an analytic function of x:

is neither non-decreasing nor non-increasing:

is not injective:

is neither non-negative nor non-positive:

has neither singularities nor discontinuities:

is neither convex nor concave:

TraditionalForm formatting:

Differentiation and Integration (5)

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:

Indefinite integral of WeberE:

More integrals:

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:

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