# SphericalBesselY

SphericalBesselY[n,z]

gives the spherical Bessel function of the second kind .

# Details • 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.

# Examples

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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(37)

### 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(12)

Real domain of :

Complex domain of : is defined for all real values greater than 0:

Complex domain is the whole plane except :

Approximate function range of :

Approximate function range of :

For integer , is an even or odd function in with the opposite parity of :

This can be expressed as :

SphericalBesselY is not an analytic function:

SphericalBesselY is neither non-decreasing nor non-increasing for non-integer n:

SphericalBesselY is not injective:

SphericalBesselY is neither non-negative nor non-positive: is singular for , possibly including , when is noninteger:

SphericalBesselY is neither convex nor concave:

### 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  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: