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
not be an integer, but must be real. The argument
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.
Y = bessely(nu,Z,1) computes
Create a column vector of domain values.
z = (0:0.2:1)';
Calculate the function values using
nu = 1.
ans = 6×1 -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)')
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 J–ν(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).
be computed using
The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,
besselh, Jν(z) is
and Yν(z) is
The Hankel functions also form a fundamental set of solutions to Bessel's
This function fully supports tall arrays. For more information, see Tall Arrays.
Usage notes and limitations:
nu must be a positive, real, integer
Z must be a positive real value.
The three-input syntax
J = bessely(nu,Z,scale) is not
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).