HankelH1

HankelH1[n,z]

gives the Hankel function of the first kind .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • is given by .
  • HankelH1[n,z] has a branch cut discontinuity in the complex z plane running from to .
  • For certain special arguments, HankelH1 automatically evaluates to exact values.
  • HankelH1 can be evaluated to arbitrary numerical precision.
  • HankelH1 automatically threads over lists.

Examples

open allclose all

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:

HankelH1 threads elementwise over lists and matrices:

Specific Values  (4)

Limiting value at infinity:

Evaluate HankelH1 symbolically for noninteger orders:

HankelH1 for symbolic n:

Find the first positive maximum for the real part of HankelH1[1/2,x]:

Visualize the result:

Visualization  (4)

Plot the absolute values and phase of the HankelH1 function for various orders:

Plot the real and imaginary parts of the HankelH1 function for various orders:

Plot the real part of TemplateBox[{0, {x, +, {i,  , y}}}, HankelH1]:

Plot the imaginary part of TemplateBox[{0, {x, +, {i,  , y}}}, HankelH1]:

Plot the real part of TemplateBox[{{1, /, 2}, {x, +, {i,  , y}}}, HankelH1]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {x, +, {i,  , y}}}, HankelH1]:

Function Properties  (4)

Complex domain for TemplateBox[{0, r}, HankelH1] and is the whole plane except :

HankelH1 is a complex linear combination of BesselJ and BesselY:

For integer and arbitrary fixed , TemplateBox[{{-, n}, z}, HankelH1]=(-1)^n TemplateBox[{n, z}, HankelH1]:

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 HankelH1:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

HankelH1 can be applied to a power series:

Properties & Relations  (2)

Use FunctionExpand to convert to Bessel functions:

Integrate expressions with HankelH1:

Possible Issues  (1)

HankelH1 does not automatically evaluate symbolically for half-integer arguments:

Use FunctionExpand to obtain an expanded form:

Introduced in 2007
 (6.0)