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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))).


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Create a column vector of domain values.

z = (0:0.2:1)';

Calculate the function values using bessely with nu = 1.

ans = 6×1


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

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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.


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).

Extended Capabilities