# BesselJ BesselJ[n,z]

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:

Elementwise threading over lists and matrices:

BesselJ can be used with Interval and CenteredInterval objects:

### 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 : 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:

### 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:

### Function Identities and Simplifications(4)

Use FullSimplify to simplify Bessel functions:

Verify the identity :

Recurrence relations :

For integer and arbitrary fixed , :

### 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: