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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.
SphericalBesselY can be used with Interval and CenteredInterval objects. »

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 (39)

Numerical Evaluation (6)

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:

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 $TemplateBox[{n, z}, SphericalBesselY]=(-1)^(n+1) TemplateBox[{n, {-, z}}, SphericalBesselY]$ :

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:

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:

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