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:

Limiting value at infinity:

SphericalBesselJ for symbolic n:

Find the first positive zero of SphericalBesselJ:

Different SphericalBesselJ types give different symbolic forms:

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 :

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 :

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:

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:

Verify the anti-derivative:

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 first kind:

Recurrence relations: