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BesselJ

BesselJ[n, z]
gives the Bessel function of the first kind J_n(z).
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • J_n(z) satisfies the differential equation z^2y^('')+zy^'+(z^2-n^2)y=0.
  • BesselJ[n, z] has a branch cut discontinuity in the complex z plane running from -infty to 0.
  • For certain special arguments, BesselJ automatically evaluates to exact values.
  • BesselJ can be evaluated to arbitrary numerical precision.
  • BesselJ automatically threads over lists.
EXAMPLESCLOSE ALL
Scope   (5)
Evaluate for complex arguments and parameters:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
BesselJ threads element-wise over lists:
For half-integer indices, BesselJ evaluates to elementary functions:
TraditionalForm formatting:
Generalizations & Extensions   (1)
BesselJ can be applied to a power series:
Applications   (2)
Solve the Bessel differential equation:
Intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle:
Properties & Relations   (3)
Use FullSimplify to simplify Bessel functions:
Sum and Integrate can produce BesselJ:
Find limits of expressions involving BesselJ:
Possible Issues   (1)
With numeric arguments, half-integer Bessel functions are not automatically evaluated:
For symbolic arguments they are:
This can lead to major inaccuracies in machine-precision evaluation:
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