|MATLAB Function Reference|
Bessel function of the first kind
The differential equation
where is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions.
and form a fundamental set of solutions of Bessel's equation for noninteger . is defined by
where is the gamma function.
is a second solution of Bessel's equation that is linearly independent of . It can be computed using
J = besselj(nu,Z)
computes the Bessel function of the first kind, , for each element of the array
Z. The order
nu need not be an integer, but must be real. The argument
Z can be complex. The result is real where
Z is positive.
Z are arrays of the same size, the result is also that size. If either input is a scalar, it is expanded to the other input's size. If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.
J = besselj(nu,Z,1)
[J,ierr] = besselj(nu,Z)
also returns completion flags in an array the same size as
||Some loss of accuracy in argument reduction.
||Unacceptable loss of accuracy,
||No convergence. Returns
The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,
besselj, and is
bessely. The Hankel functions also form a fundamental set of solutions to Bessel's equation (see
besselj(3:9,(0:.2:10)') generates the entire table on page 398 of  Abramowitz and Stegun, Handbook of Mathematical Functions.
besselj function uses a Fortran MEX-file to call a library developed by D. E. Amos  .
 Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sections 9.1.1, 9.1.89 and 9.12, formulas 9.1.10 and 9.2.5.
 Carrier, Krook, and Pearson, Functions of a Complex Variable: Theory and Technique, Hod Books, 1983, section 5.5.
 Amos, D. E., "A Subroutine Package for Bessel Functions of a Complex Argument and Nonnegative Order," Sandia National Laboratory Report, SAND85-1018, May, 1985.
 Amos, D. E., "A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order," Trans. Math. Software, 1986.