# BesselJZero

BesselJZero[n,k]

represents the k zero of the Bessel function .

BesselJZero[n,k,x0]

represents the k zero greater than x0.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• N[BesselJZero[n,k]] gives a numerical approximation so long as the specified zero exists.
• BesselJZero[n,k] represents the k zero greater than 0.
• BesselJZero can be evaluated to arbitrary numerical precision.
• BesselJZero automatically threads over lists. »

# Examples

open allclose all

## Basic Examples(5)

Evaluate numerically:

Evaluate symbolically:

Display zeros of the BesselJ function over a subset of the reals:

Series expansion at the origin:

## Scope(18)

### Numerical Evaluation(7)

Evaluate numerically:

Find the first zero of greater than 40:

Evaluate to high precision:

Evaluate efficiently at high precision:

Evaluate at a non-integer second argument:

For BesselJZero[ν,k-α/π], the result is a zero of :

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix BesselJZero function using MatrixFunction:

### Specific Values(3)

Limiting value at infinity:

The first three zeros:

Find the first zero of BesselJ[1,x] using Solve:

### Visualization(3)

Visualize the zeroes of BesselJ as a step function:

Display zeros of the BesselJ function:

Show the first zero greater than 6:

### Differentiation and Series Expansions(5)

Find the derivative of Bessel zero with respect to k:

Second derivative:

Find the Taylor expansion using Series:

Find the series expansion at Infinity:

Taylor expansion at a generic point:

## Applications(3)

Find the first 10 eigenmodes of a circular drum with Dirichlet boundary conditions:

Construct an amplitude comprising a certain mixture of modes:

Circular density plot:

Find the coefficient in the Rayleigh criterion for diffraction-limited optics:

Analytically compute the eigenvalues of a Laplacian in Cartesian coordinates over a Disk:

## Properties & Relations(1)

Asymptotic behavior of BesselJZero[ν,k] for large k:

Wolfram Research (2007), BesselJZero, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselJZero.html.

#### Text

Wolfram Research (2007), BesselJZero, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselJZero.html.

#### CMS

Wolfram Language. 2007. "BesselJZero." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BesselJZero.html.

#### APA

Wolfram Language. (2007). BesselJZero. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselJZero.html

#### BibTeX

@misc{reference.wolfram_2024_besseljzero, author="Wolfram Research", title="{BesselJZero}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/BesselJZero.html}", note=[Accessed: 06-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_besseljzero, organization={Wolfram Research}, title={BesselJZero}, year={2007}, url={https://reference.wolfram.com/language/ref/BesselJZero.html}, note=[Accessed: 06-August-2024 ]}