BesselJZero

BesselJZero[n,k]

represents the k^(th) zero of the Bessel function .

BesselJZero[n,k,x0]

represents the k^(th) 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^(th) zero greater than 0.
  • BesselJZero can be evaluated to arbitrary numerical precision.
  • BesselJZero automatically threads over lists. »

Examples

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Basic Examples  (5)

Evaluate numerically:

Evaluate symbolically:

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

Series expansion at the origin:

TraditionalForm formatting:

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:

Radial drum displacement profile:

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_2025_besseljzero, author="Wolfram Research", title="{BesselJZero}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/BesselJZero.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_besseljzero, organization={Wolfram Research}, title={BesselJZero}, year={2007}, url={https://reference.wolfram.com/language/ref/BesselJZero.html}, note=[Accessed: 26-March-2025 ]}