This is machine translation

Translated by Microsoft
Mouse over text to see original. Click the button below to return to the English verison of the page.


Bessel function of second kind


Y = bessely(nu,Z)
Y = bessely(nu,Z,1)


Y = bessely(nu,Z) computes Bessel functions of the second kind, Yν(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.

Y = bessely(nu,Z,1) computes bessely(nu,Z).*exp(-abs(imag(Z))).


collapse all

Create a column vector of domain values.

z = (0:0.2:1)';

Calculate the function values using bessely with nu = 1.

ans =


Define the domain.

X = 0:0.1:20;

Calculate the first five Bessel functions of the second kind.

Y = zeros(5,201);
for i = 0:4
    Y(i+1,:) = bessely(i,X);

Plot the results.

axis([-0.1 20.2 -2 0.6])
grid on
title('Bessel Functions of the Second Kind for v = 0,1,2,3,4')

More About

collapse all

Bessel's Equation

The differential equation


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

A solution Yν(z) of the second kind can be expressed as


where Jν(z) and Jν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν


and Γ(a) is the gamma function. Yν(z) is linearly independent of Jν(z).

Jν(z) can be computed using besselj.

Tall Array Support

This function fully supports tall arrays. For more information, see Tall Arrays.


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


where Hν(K)(z) 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).

Was this topic helpful?