# SphericalBesselJ

SphericalBesselJ[n,z]

gives the spherical Bessel function of the first kind .

# Examples

open allclose all

## 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(38)

### Numerical Evaluation(5)

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:

SphericalBesselJ can be used with Interval and CenteredInterval objects:

### 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 :

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:

Wolfram Research (2007), SphericalBesselJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalBesselJ.html.

#### Text

Wolfram Research (2007), SphericalBesselJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalBesselJ.html.

#### CMS

Wolfram Language. 2007. "SphericalBesselJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalBesselJ.html.

#### APA

Wolfram Language. (2007). SphericalBesselJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalBesselJ.html

#### BibTeX

@misc{reference.wolfram_2022_sphericalbesselj, author="Wolfram Research", title="{SphericalBesselJ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalBesselJ.html}", note=[Accessed: 20-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_sphericalbesselj, organization={Wolfram Research}, title={SphericalBesselJ}, year={2007}, url={https://reference.wolfram.com/language/ref/SphericalBesselJ.html}, note=[Accessed: 20-March-2023 ]}