bessely

Bessel function of the second kind for symbolic expressions

Syntax

bessely(nu,z)

Description

bessely(nu,z) returns the Bessel function of the second kind, Y_ν(z).

Examples

Find Bessel Function of Second Kind

Compute the Bessel functions of the second kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

[bessely(0, 5), bessely(-1, 2), bessely(1/3, 7/4), bessely(1, 3/2 + 2*i)]

ans =
  -0.3085 + 0.0000i   0.1070 + 0.0000i   0.2358 + 0.0000i  -0.4706 + 1.5873i

Compute the Bessel functions of the second kind for the numbers converted to symbolic objects. For most symbolic (exact) numbers, bessely returns unresolved symbolic calls.

[bessely(sym(0), 5), bessely(sym(-1), 2),...
 bessely(1/3, sym(7/4)), bessely(sym(1), 3/2 + 2*i)]

ans =
[ bessely(0, 5), -bessely(1, 2), bessely(1/3, 7/4), bessely(1, 3/2 + 2i)]

For symbolic variables and expressions, bessely also returns unresolved symbolic calls:

syms x y
[bessely(x, y), bessely(1, x^2), bessely(2, x - y), bessely(x^2, x*y)]

ans =
[ bessely(x, y), bessely(1, x^2), bessely(2, x - y), bessely(x^2, x*y)]

Solve Bessel Differential Equation for Bessel Functions

Solve this second-order differential equation. The solutions are the Bessel functions of the first and the second kind.

syms nu w(z)
dsolve(z^2*diff(w, 2) + z*diff(w) +(z^2 - nu^2)*w == 0)

ans =
C2*besselj(nu, z) + C3*bessely(nu, z)

Verify that the Bessel function of the second kind is a valid solution of the Bessel differential equation:

syms nu z
isAlways(z^2*diff(bessely(nu, z), z, 2) + z*diff(bessely(nu, z), z)...
 + (z^2 - nu^2)*bessely(nu, z) == 0)

ans =
  logical
   1

Special Values of Bessel Function of Second Kind

If the first parameter is an odd integer multiplied by 1/2, bessely rewrites the Bessel functions in terms of elementary functions:

syms x
bessely(1/2, x)

ans =
-(2^(1/2)*cos(x))/(x^(1/2)*pi^(1/2))

bessely(-1/2, x)

ans =
(2^(1/2)*sin(x))/(x^(1/2)*pi^(1/2))

bessely(-3/2, x)

ans =
(2^(1/2)*(cos(x) - sin(x)/x))/(x^(1/2)*pi^(1/2))

bessely(5/2, x)

ans =
-(2^(1/2)*((3*sin(x))/x + cos(x)*(3/x^2 - 1)))/(x^(1/2)*pi^(1/2))

Differentiate Bessel Functions of Second Kind

Differentiate the expressions involving the Bessel functions of the second kind:

syms x y
diff(bessely(1, x))
diff(diff(bessely(0, x^2 + x*y -y^2), x), y)

ans =
bessely(0, x) - bessely(1, x)/x
 
ans =
- bessely(1, x^2 + x*y - y^2) -...
(2*x + y)*(bessely(0, x^2 + x*y - y^2)*(x - 2*y) -...
(bessely(1, x^2 + x*y - y^2)*(x - 2*y))/(x^2 + x*y - y^2))

Find Bessel Function for Matrix Input

Call bessely for the matrix A and the value 1/2. The result is a matrix of the Bessel functions bessely(1/2, A(i,j)).

syms x
A = [-1, pi; x, 0];
bessely(1/2, A)

ans =
[         (2^(1/2)*cos(1)*1i)/pi^(1/2), 2^(1/2)/pi]
[ -(2^(1/2)*cos(x))/(x^(1/2)*pi^(1/2)),        Inf]

Plot Bessel Functions of Second Kind

Plot the Bessel functions of the second kind for $$v=0,1,2,3$$.

syms x y
fplot(bessely(0:3,x))
axis([0 10 -1 0.6])
grid on

ylabel('Y_v(x)')
legend('Y_0','Y_1','Y_2','Y_3', 'Location','Best')
title('Bessel functions of the second kind')

Input Arguments

nu — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

If nu is a vector or matrix, bessely returns the Bessel function of the second kind for each element of nu.

z — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

If z is a vector or matrix, bessely returns the Bessel function of the second kind for each element of z.

More About

Bessel Function of the Second Kind

The Bessel differential equation

$$z^2\frac{d^{2}w}{d{z}^2}+z\frac{dw}{dz}+\left(z^2-{\nu }^2\right)w=0$$

has two linearly independent solutions. These solutions are represented by the Bessel functions of the first kind, J_ν(z), and the Bessel functions of the second kind, Y_ν(z):

$$w\left(z\right)=C_1{J}_{\nu }\left(z\right)+C_2{Y}_{\nu }\left(z\right)$$

The Bessel functions of the second kind are defined via the Bessel functions of the first kind:

$$Y_{\nu }\left(z\right)=\frac{J_{\nu }\left(z\right)\cos \left(\nu \pi \right)-J_{-\nu }\left(z\right)}{\sin \left(\nu \pi \right)}$$

Here J_ν(z) are the Bessel function of the first kind:

$$J_{\nu }\left(z\right)=\frac{{\left(z/2\right)}^{\nu }}{\sqrt{\pi }\Gamma \left(\nu +1/2\right)}{\displaystyle \underset{0}{\overset{\pi }{\int }}\cos \left(z\cos \left(t\right)\right)\sin {\left(t\right)}^{2\nu }dt}$$

Tips

• Calling bessely for a number that is not a symbolic object invokes the MATLAB^® bessely function.

  At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, bessely(nu,z) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

References

[1] Olver, F. W. J. “Bessel Functions of Integer Order.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

[2] Antosiewicz, H. A. “Bessel Functions of Fractional Order.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

Version History

Introduced in R2014a