# 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

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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(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 :

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: 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:

### 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: