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# BesselJ

 BesselJgives the Bessel function of the first kind .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• satisfies the differential equation .
• BesselJ has a branch cut discontinuity in the complex z plane running from to .
• For certain special arguments, BesselJ automatically evaluates to exact values.
• BesselJ can be evaluated to arbitrary numerical precision.
• BesselJ automatically threads over lists.
Evaluate numerically:
Plot :
Evaluate numerically:
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Plot :
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 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:
BesselJ can be applied to a power series:
 Applications   (3)
Solve the Bessel differential equation:
Intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle:
Approximate solution of Kepler's equation as a truncated Fourier sine series:
Exact solution:
Plot the difference between solutions:
Use FullSimplify to simplify Bessel functions:
Sum and Integrate can produce BesselJ:
Find limits of expressions involving BesselJ:
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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