gives the spherical Bessel function of the second kind .


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


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (32)

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:

SphericalBesselY for symbolic n:

Find the first positive zero of SphericalBesselY:

Different SphericalBesselY types give different symbolic forms:

Visualization  (3)

Plot the SphericalBesselY function for integer () and half-integer () orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

Function Properties  (7)

Real domain of TemplateBox[{0, x}, SphericalBesselY]:

Complex domain of TemplateBox[{0, x}, SphericalBesselY]:

TemplateBox[{{-, {1, /, 2}}, x}, SphericalBesselY] is defined for all real values greater than 0:

Complex domain is the whole plane except :

Approximate function range of TemplateBox[{0, x}, SphericalBesselY]:

Approximate function range of TemplateBox[{1, x}, SphericalBesselY]:

TemplateBox[{0, x}, SphericalBesselY] is an odd function:

TemplateBox[{1, x}, SphericalBesselY] is an even function:

SphericalBesselY threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z

Plot the higher derivatives 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  (6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:


Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Use FullSimplify to simplify spherical Bessel functions of the second kind:

Recurrence relations:

Applications  (1)

Solve the radial part of the three-dimensional Laplace operator:

Properties & Relations  (1)

Integrate expressions involving spherical Bessel functions:

Introduced in 2007