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SphericalBesselJ

SphericalBesselJ[n,z]

gives the spherical Bessel function of the first kind .

Details

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

Specific Values (4)

Limiting value at infinity:

SphericalBesselJ for symbolic n:

Find the first positive zero of SphericalBesselJ:

Different SphericalBesselJ types give different symbolic forms:

Visualization (3)

Plot the SphericalBesselJ 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)

is defined for all real and complex values:

is defined for all real values greater than 0:

Complex domain is the whole plane except :

Approximate function range of :

For integer , is an even or odd function in depending on whether is even or odd:

This can be expressed as $TemplateBox[{n, z}, BesselJ]=(-1)^n TemplateBox[{n, {-, z}}, BesselJ]$ :

SphericalBesselJ threads elementwise over lists:

is not an analytic function of for noninteger and negative values of :

SphericalBesselJ is neither non-decreasing nor non-increasing:

SphericalBesselJ is not injective:

SphericalBesselJ is neither non-negative nor non-positive:

is singular for , possibly including , when is noninteger:

SphericalBesselJ 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:

Verify the anti-derivative:

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 first kind:

Recurrence relations:

Applications (1)

Solve the radial part of the Laplace operator in 3D:

Properties & Relations (2)

SphericalBesselJ can be represented as a DifferentialRoot:

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