# 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

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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:

## 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 :

### 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: