Transcendental equation zj1=cj0 involving Bessel functions

Find N non-negative roots of transcendental equation z*J1(z) - c*J0(z) = 0, where J1(z) and J0(z) are Bessel functions of the first kind.

ZJ1CJ0 - Find the first N non-negative roots of the transcendental equation z*J1(z) - c*J0(z) = 0, where J1(z) and J0(z) are Bessel functions of the first kind of order 1 and 0, respectively.

Syntax:

• roots = zj1cj0(c, N)
• roots = zj1cj0(c) % Default N: 10

Description:

This function finds the first N non-negative roots of the transcendental equation z*J1(z) = c*J0(z), where J0 and J1 are Bessel functions of the first kind.

Inputs:

• c - A non-negative constant, typical the Biot number Bi in heat transder or the Sherwood number Sh in mass transfer.
• N - (Optional) Number of positive roots to find. Default is 10.

Outputs:

• roots - A 1xN array of the first N non-negative roots

Example:

• roots = zj1cj0(2, 5) % Finds 5 roots when c = 2
• roots = zj1cj0(2) % Finds 10 roots when c = 2 (default N)

See also: besselj, fzero

Copyright © 2025 Jie Zhang (https://orcid.org/0000-0002-8023-8144)