SphericalHankelH1

SphericalHankelH1[n,z]

gives the spherical Hankel function of the first kind .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • SphericalHankelH1 is given in terms of ordinary Hankel functions by .
  • SphericalHankelH1[n,z] has a branch cut discontinuity in the complex z plane running from to .
  • Explicit symbolic forms for integer n can be obtained using FunctionExpand.
  • For certain special arguments, SphericalHankelH1 automatically evaluates to exact values.
  • SphericalHankelH1 can be evaluated to arbitrary numerical precision.
  • SphericalHankelH1 automatically threads over lists.

Examples

open allclose all

Basic Examples  (6)

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:

Series expansion at a singular point:

Scope  (30)

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  (4)

Limiting value at infinity:

SphericalHankelH1 for symbolic n:

Find the first positive zero of imaginary part of SphericalHankelH1:

Different SphericalHankelH1 types give different symbolic forms:

Visualization  (3)

Plot the absolute values of SphericalHankelH1 function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

Function Properties  (7)

Complex domain for TemplateBox[{n, z}, SphericalHankelH1] is the whole plane except :

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

SphericalHankelH1 is a complex linear combination of SphericalBesselJ and SphericalBesselY:

SphericalHankelH1 threads elementwise over lists:

TemplateBox[{n, x}, SphericalHankelH1] is not an analytic function:

SphericalHankelH1 is not injective over complexes:

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

TemplateBox[{n, z}, SphericalHankelH1] has both singularities and discontinuities along the non-positive real axis:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the absolute values of the higher derivatives of with respect to :

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

Integration  (3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Use FullSimplify to simplify spherical Hankel functions of the first kind:

Recurrence relations:

Properties & Relations  (1)

Wolfram Research (2007), SphericalHankelH1, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHankelH1.html.

Text

Wolfram Research (2007), SphericalHankelH1, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHankelH1.html.

BibTeX

@misc{reference.wolfram_2021_sphericalhankelh1, author="Wolfram Research", title="{SphericalHankelH1}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalHankelH1.html}", note=[Accessed: 22-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_sphericalhankelh1, organization={Wolfram Research}, title={SphericalHankelH1}, year={2007}, url={https://reference.wolfram.com/language/ref/SphericalHankelH1.html}, note=[Accessed: 22-September-2021 ]}

CMS

Wolfram Language. 2007. "SphericalHankelH1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalHankelH1.html.

APA

Wolfram Language. (2007). SphericalHankelH1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalHankelH1.html