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# besselj

Bessel function of first kind

## Syntax

J = besselj(nu,Z)
J = besselj(nu,Z,1)

## Definitions

The differential equation

${z}^{2}\frac{{d}^{2}y}{d{z}^{2}}+z\frac{dy}{dz}+\left({z}^{2}-{\nu }^{2}\right)y=0,$

where ν is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions.

Jν(z) and Jν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν. Jν(z) is defined by

${J}_{\nu }\left(z\right)={\left(\frac{z}{2}\right)}^{\nu }\sum _{\left(k=0\right)}^{\infty }\frac{{\left(\frac{-{z}^{2}}{4}\right)}^{k}}{k!\Gamma \left(\nu +k+1\right)}$

where Γ(a) is the gamma function.

Yν(z) is a second solution of Bessel's equation that is linearly independent of Jν(z). It can be computed using bessely.

## Description

J = besselj(nu,Z) computes the Bessel function of the first kind, Jν(z), 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.

If nu and 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.

J = besselj(nu,Z,1) computes besselj(nu,Z).*exp(-abs(imag(Z))).

## Examples

expand all

### Vector of Function Values

Create a column vector of domain values.

z = (0:0.2:1)';

Calculate the function values using besselj with nu = 1.

format long
besselj(1,z)
ans =

0
0.099500832639236
0.196026577955319
0.286700988063916
0.368842046094170
0.440050585744934

### Plot Bessel Functions of First Kind

Define the domain.

X = 0:0.1:20;

Calculate the first five Bessel functions of the first kind.

J = zeros(5,201);
for i = 0:4
J(i+1,:) = besselj(i,X);
end

Plot the results.

plot(X,J,'LineWidth',1.5)
axis([0 20 -.5 1])
grid on
legend('J_0','J_1','J_2','J_3','J_4','Location','Best')
title('Bessel Functions of the First Kind for v = 0,1,2,3,4')
xlabel('X')
ylabel('J_v(X)')

expand all

### Tips

The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,

$\begin{array}{l}{H}_{\nu }^{\left(1\right)}\left(z\right)={J}_{\nu }\left(z\right)+i{Y}_{\nu }\left(z\right)\\ {H}_{\nu }^{\left(2\right)}\left(z\right)={J}_{\nu }\left(z\right)-i{Y}_{\nu }\left(z\right)\end{array}$

where ${H}_{\nu }^{\left(K\right)}\left(z\right)$is besselh, Jν(z) is besselj, and Yν(z) is bessely. The Hankel functions also form a fundamental set of solutions to Bessel's equation (see besselh).

### Algorithms

The besselj function uses a Fortran MEX-file to call a library developed by D.E. Amos [3] [4].

## References

[1] 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.

[2] Carrier, Krook, and Pearson, Functions of a Complex Variable: Theory and Technique, Hod Books, 1983, section 5.5.

[3] Amos, D.E., "A Subroutine Package for Bessel Functions of a Complex Argument and Nonnegative Order," Sandia National Laboratory Report, SAND85-1018, May, 1985.

[4] Amos, D.E., "A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order," Trans. Math. Software, 1986.