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bessely

Bessel function of second kind

Syntax

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

Description

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))).

Examples

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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.

bessely(1,z)
ans =

      -Inf
   -3.3238
   -1.7809
   -1.2604
   -0.9781
   -0.7812

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);
end

Plot the results.

plot(X,Y,'LineWidth',1.5)
axis([-0.1 20.2 -2 0.6])
grid on
legend('Y_0','Y_1','Y_2','Y_3','Y_4','Location','Best')
title('Bessel Functions of the Second Kind for v = 0,1,2,3,4')
xlabel('X')
ylabel('Y_v(X)')

More About

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Bessel's Equation

The differential equation

z2d2ydz2+zdydz+(z2ν2)y=0,

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

Yν(z)=Jν(z)cos(νπ)Jν(z)sin(νπ)

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

Jv(z)=(z2)νk=0(z24)kk!Γ(ν+k+1),

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.

Tips

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

Hν(1)(z)=Jν(z)+iYν(z)Hν(2)(z)=Jν(z)iYν(z),

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).

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