gives the spherical Bessel function of the first kind .
- 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. »
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
SphericalBesselJ can be used with Interval and CenteredInterval objects:
Specific Values (4)
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:
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:
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:
Compute the indefinite integral using Integrate:
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 :
Function Identities and Simplifications (2)
Use FullSimplify to simplify spherical Bessel functions of the first kind:
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.
Wolfram Language. 2007. "SphericalBesselJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalBesselJ.html.
Wolfram Language. (2007). SphericalBesselJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalBesselJ.html