gives the spherical Hankel function of the first kind .


  • 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.


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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  (28)

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  (5)

Complex domain for TemplateBox[{0, z}, SphericalHankelH1] and TemplateBox[{{-, {1, /, 2}}, z}, SphericalHankelH1] is the whole plane except :

TemplateBox[{0, z}, SphericalHankelH1] is an even function:

TemplateBox[{1, z}, SphericalHankelH1] is an odd function:

SphericalHankelH1 threads elementwise over lists:

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)

Introduced in 2007