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BesselJ

See Also
- BesselJZero
- BesselY
- SphericalBesselJ
- HankelH1
- HankelH2
- KelvinBer
- KelvinBei
- StruveH
- BesselK
- BesselI
- AngerJ
- WeberE
- StruveL
- ZernikeR
- HeunD
- HankelTransform
Related Guides
Tech Notes
- Special Functions
- Some Notes on Internal Implementation

BesselJ

gives the Bessel function of the first kind .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
satisfies the differential equation .
BesselJ[n,z] 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.
BesselJ can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (52)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments and parameters:

Evaluate BesselJ efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix BesselJ function using MatrixFunction:

Specific Values (3)

For half-integer orders, BesselJ evaluates to elementary functions:

Limiting value at infinity:

The first three zeros of :

Find the first positive zero of using Solve:

Visualize the result:

Visualization (4)

Plot the BesselJ function for integer () and half-integer () orders:

Plot the real and imaginary parts of the BesselJ function for half-integer orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

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 :

Approximate function range of :

For integer , is an even or odd function in depending on whether is even or odd:

This can be expressed as $TemplateBox[{n, z}, BesselJ]=(-1)^n TemplateBox[{n, {-, z}}, BesselJ]$ :

is an analytic function of for integer :

It is not analytic for noninteger orders:

BesselJ is neither non-decreasing nor non-increasing:

BesselJ is not injective:

BesselJ is not surjective:

BesselJ is neither non-negative nor non-positive:

is singular for , possibly including , when is noninteger:

The same is true of its discontinuities:

BesselJ is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for integer and half-integer orders:

Formula for the derivative:

Integration (5)

Compute the indefinite integral of BesselJ using Integrate:

Indefinite integral of an expression involving BesselJ:

Definite integral:

Definite integral of over an interval centered at the origin is 0:

Definite integral of (even integrand) over an interval centered at the origin:

This is twice the integral over half the interval:

Series Expansions (6)

Taylor expansion for around :

Plot the first three approximations for around :

General term in the series expansion of BesselJ:

Series expansion for around :

Plot the first three approximations for around :

Asymptotic approximation of BesselJ:

Taylor expansion at a generic point:

BesselJ can be applied to a power series:

Integral Transforms (4)

Compute a Fourier transform using FourierTransform:

LaplaceTransform:

HankelTransform:

MellinTransform:

Function Identities and Simplifications (4)

Use FullSimplify to simplify Bessel functions:

Verify the identity :

Recurrence relations $z (TemplateBox[{{n, -, 1}, z}, BesselJ] + TemplateBox[{{n, +, 1}, z}, BesselJ])=2 n J_n(z)$ :

For integer and arbitrary fixed , $TemplateBox[{{-, n}, z}, BesselJ]=(-1)^n TemplateBox[{n, z}, BesselJ]$ :

Function Representations (5)

Representation through BesselI:

Series representation:

Integral representation:

Representation in terms of MeijerG:

Representation in terms of DifferenceRoot:

Applications (3)

Solve the Bessel differential equation:

Solve another differential equation:

Fraunhofer diffraction is the type of diffraction that occurs in the limit of a small Fresnel number. Plot the intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle:

Kepler's equation describes the motion of a body in an elliptical orbit. Approximate solution of Kepler's equation as a truncated Fourier sine series:

Exact solution:

Plot the difference between solutions:

Properties & Relations (5)

Use FullSimplify to simplify Bessel functions:

Sum and Integrate can produce BesselJ:

Find limits of expressions involving BesselJ:

BesselJ can be represented as a DifferentialRoot:

The exponential generating function for 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:

Neat Examples (1)

Plot the Riemann surface of :

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