HankelH2

HankelH2[n,z]

gives the Hankel function of the second kind .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • is given by .
  • HankelH2[n,z] has a branch cut discontinuity in the complex z plane running from to .
  • For certain special arguments, HankelH2 automatically evaluates to exact values.
  • HankelH2 can be evaluated to arbitrary numerical precision.
  • HankelH2 automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot the real and imaginary parts of the function:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (29)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

HankelH2 threads elementwise over lists and matrices:

Specific Values  (4)

Limiting value at infinity:

Evaluate HankelH2 symbolically for noninteger orders:

HankelH2 for symbolic n:

Find the first positive minimum for the imaginary part of HankelH2[1/2,x]:

Visualize the result:

Visualization  (4)

Plot the absolute values and phase of the HankelH2 function for various orders:

Plot the real and imaginary parts of the HankelH2 function for various orders:

Plot the real part of TemplateBox[{0, {x, +, {i,  , y}}}, HankelH2]:

Plot the imaginary part of TemplateBox[{0, {x, +, {i,  , y}}}, HankelH2]:

Plot the real part of TemplateBox[{{1, /, 2}, {x, +, {i,  , y}}}, HankelH2]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {x, +, {i,  , y}}}, HankelH2]:

Function Properties  (4)

Complex domain for TemplateBox[{0, z}, HankelH2] and is the whole plane except :

HankelH2 is a linear combination of BesselJ and BesselY:

For integer and arbitrary fixed , TemplateBox[{{-, n}, z}, HankelH2]=(-1)^n TemplateBox[{n, z}, HankelH2]:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=2:

Formula for the ^(th) derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

Series Expansions  (6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Asymptotic approximation of HankelH2:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

HankelH2 can be applied to a power series:

Properties & Relations  (2)

Use FunctionExpand to convert to Bessel functions:

Integrate expressions with HankelH2:

Possible Issues  (1)

HankelH2 does not automatically evaluate symbolically for half-integer arguments:

Use FunctionExpand to obtain expanded form:

Introduced in 2007
 (6.0)