BUILT-IN MATHEMATICA SYMBOL

BesselJ

gives the Bessel function of the first kind

.

MORE INFORMATION

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 .

FullSimplify and FunctionExpand include transformation rules for BesselJ.

For certain special arguments, BesselJ automatically evaluates to exact values.

BesselJ can be evaluated to arbitrary numerical precision.

BesselJ automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Plot

:

Evaluate numerically:

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Plot

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Out[1]=

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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:

TraditionalForm formatting:

Generalizations & Extensions (1)

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:

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: