gives the spherical Hankel function of the second kind .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • SphericalHankelH2 is given in terms of ordinary Hankel functions by .
  • SphericalHankelH2[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, SphericalHankelH2 automatically evaluates to exact values.
  • SphericalHankelH2 can be evaluated to arbitrary numerical precision.
  • SphericalHankelH2 automatically threads over lists.


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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:

SphericalHankelH2 for symbolic n:

Find the first positive zero of imaginary part of SphericalHankelH2:

Different SphericalHankelH2 types give different symbolic forms:

Visualization  (3)

Plot the absolute values of SphericalHankelH2 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}, SphericalHankelH2] is the whole plane except :

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

SphericalHankelH2 is a complex linear combination of SphericalBesselJ and SphericalBesselY:

SphericalHankelH2 threads elementwise over lists:

TemplateBox[{n, z}, SphericalHankelH2] is not an analytic function:

SphericalHankelH2 is not injective over complexes:

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

TemplateBox[{n, z}, SphericalHankelH2] has both singularities and discontinuities along the non-positive real axis:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z:

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

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 second kind:

Recurrence relations:

Properties & Relations  (1)

Wolfram Research (2007), SphericalHankelH2, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHankelH2.html.


Wolfram Research (2007), SphericalHankelH2, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalHankelH2.html.


@misc{reference.wolfram_2021_sphericalhankelh2, author="Wolfram Research", title="{SphericalHankelH2}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalHankelH2.html}", note=[Accessed: 22-September-2021 ]}


@online{reference.wolfram_2021_sphericalhankelh2, organization={Wolfram Research}, title={SphericalHankelH2}, year={2007}, url={https://reference.wolfram.com/language/ref/SphericalHankelH2.html}, note=[Accessed: 22-September-2021 ]}


Wolfram Language. 2007. "SphericalHankelH2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalHankelH2.html.


Wolfram Language. (2007). SphericalHankelH2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalHankelH2.html