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  (17)

Numerical Evaluation  (6)

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 :

BesselJZero threads elementwise over lists:

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  (2)

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:

Properties & Relations  (1)

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

Introduced in 2007
 (6.0)