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BesselJ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

BesselJ[n, z]
    gives the Bessel function of the first kind J_n(z), a solution of
    z^2 y'' + z y' + (z^2 - n^2) y = 0 regular at the origin.
J_0(0) = 1, J_n(0) = 0 for integer n != 0. Has a branch cut along the
negative real z axis for non-integer n. Real and complex order and
argument evaluate numerically at machine or arbitrary (MPFR) precision;
D[BesselJ[n, z], z] = (BesselJ[n-1, z] - BesselJ[n+1, z])/2. Listable.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= BesselJ[0, 5.2]
Out[1]= -0.11029

In[2]:= D[BesselJ[n, x], x]
Out[2]= 1/2 (BesselJ[-1 + n, x] - BesselJ[1 + n, x])

Implementation notes

Attributes: Listable, NumericFunction, Protected, ReadProtected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= BesselJ[0, 0]
Out[1]= 1

In[2]:= BesselJ[1, 0]
Out[2]= 0

Half-integer orders close in elementary functions:

In[1]:= BesselJ[1/2, z]
Out[1]= Sin[z] Sqrt[2/(Pi z)]

The Frobenius series at the origin:

In[1]:= Series[BesselJ[0, x], {x, 0, 6}]
Out[1]= 1 - 1/4 x^2 + 1/64 x^4 - 1/2304 x^6 + O[x]^7

High-precision and complex evaluation; the first input is J_0 at its first zero, which numerically returns essentially zero:

In[1]:= N[BesselJ[0, 1], 40]
Out[1]= 0.7651976865579665514497175261026632209093

In[2]:= N[BesselJ[0, 10 + 5 I], 30]
Out[2]= -17.78959112945037151834426180967 + 0.2007116167212048509818027697064*I

Notes

BesselJ[n, z] is the Bessel function of the first kind, regular at the origin, with J_0(0) = 1 and J_n(0) = 0 for integer n != 0. Real and complex order and argument evaluate at machine or MPFR precision; D[BesselJ[n, z], z] = (BesselJ[n-1, z] - BesselJ[n+1, z])/2. There is a branch cut along the negative real axis for non-integer order. Listable.