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.
  • HankelH2 can be used with a CenteredInterval object. »

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  (33)

Numerical Evaluation  (6)

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:

HankelH2 can be used with a CenteredInterval object:

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  (7)

Complex domain for TemplateBox[{n, r}, HankelH2] is the whole plane except :

It is not defined as a function from TemplateBox[{}, Reals] to TemplateBox[{}, Reals]:

HankelH2 is a linear combination of BesselJ and BesselY:

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

TemplateBox[{n, z}, HankelH2] is not an analytic function of :

HankelH2 is not injective over complexes:

Use FindInstance to find inputs that demonstrate it is not injective:

TemplateBox[{n, z}, HankelH2] has both singularities and discontinuities along the negative real axis:

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:

Applications  (1)

There can be subtleties with asymptotic approximation when the function to be approximated approaches zero infinitely many times in every neighborhood of the approximation point. As an example, consider the asymptotic expansion of TemplateBox[{1, x}, BesselJ] near :

Consider the approximation of the never-zero Hankel function TemplateBox[{1, x}, HankelH1]:

This approximation is asymptotic:

So is the approximation of the Hankel function of the second kind, TemplateBox[{1, x}, HankelH2]:

As TemplateBox[{1, x}, BesselJ]=1/2 (TemplateBox[{1, x}, HankelH1]+TemplateBox[{1, x}, HankelH2]), its approximation can be understood as nearly asymptotic, being the sum
of two such approximations:

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:

Wolfram Research (2007), HankelH2, Wolfram Language function, https://reference.wolfram.com/language/ref/HankelH2.html.

Text

Wolfram Research (2007), HankelH2, Wolfram Language function, https://reference.wolfram.com/language/ref/HankelH2.html.

CMS

Wolfram Language. 2007. "HankelH2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HankelH2.html.

APA

Wolfram Language. (2007). HankelH2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HankelH2.html

BibTeX

@misc{reference.wolfram_2022_hankelh2, author="Wolfram Research", title="{HankelH2}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/HankelH2.html}", note=[Accessed: 31-January-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_hankelh2, organization={Wolfram Research}, title={HankelH2}, year={2007}, url={https://reference.wolfram.com/language/ref/HankelH2.html}, note=[Accessed: 31-January-2023 ]}