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:

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:

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 :

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:

TraditionalForm formatting:

First derivative:

Higher derivatives:

Plot higher derivatives for integer and half-integer orders:

Formula for the derivative:

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:

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:

Compute a Fourier transform using FourierTransform:

LaplaceTransform:

HankelTransform:

MellinTransform:

Use FullSimplify to simplify Bessel functions:

Verify the identity :

Recurrence relations :

For integer and arbitrary fixed , :

Representation through BesselI:

Series representation:

Integral representation:

Representation in terms of MeijerG:

Representation in terms of DifferenceRoot: