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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix SphericalBesselY function using MatrixFunction:

Limiting value at infinity:

SphericalBesselY for symbolic n:

Find the first positive zero of SphericalBesselY:

Different SphericalBesselY types give different symbolic forms:

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 :

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 threads elementwise over lists:

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:

TraditionalForm formatting:

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:

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

FourierSeries:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

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

Recurrence relations: