贝斯ly

Bessel function of the second kind for symbolic expressions

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Syntax

贝斯ly(nu,z)

Description

example

贝斯ly(nu,z)returns theBessel 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,贝斯lyreturns unresolved symbolic calls.

(贝斯(信谊(0),5),贝斯(信谊(1),2),…贝斯ly(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,贝斯lyalso 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,贝斯lyrewrites 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))

贝斯ly(-1/2, x)

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

贝斯ly(-3/2, x)

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

贝斯ly(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贝斯lyfor the matrixAand the value 1/2. The result is a matrix of the Bessel functions贝斯ly(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 $v = 0, 1, 2, 3$ .

symsxyfplot(bessely(0:3,x)) axis([0 10 -1 0.6]) gridonylabel('Y_v(x)') legend('Y_0','Y_1','Y_2','Y_3','Location','Best') title('Bessel functions of the second kind')

Figure contains an axes object. The axes object with title Bessel functions of the second kind contains 4 objects of type functionline. These objects represent Y_0, Y_1, Y_2, Y_3.

Input Arguments

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`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.

Ifnuis a vector or matrix,贝斯lyreturns the Bessel function of the second kind for each element ofnu.

`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.

Ifzis a vector or matrix,贝斯lyreturns the Bessel function of the second kind for each element ofz.

More About

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Bessel Function of the Second Kind

贝塞尔微分equation

$z^{2} \frac{d^{2} w}{d z^{2}} + z \frac{d w}{d z} + (z^{2} - ν^{2}) w = 0$

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):

$w (z) = C_{1} J_{ν} (z) + C_{2} Y_{ν} (z)$

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

$Y_{ν} (z) = \frac{J_{ν} (z) \cos (ν π) - J_{- ν} (z)}{\sin (ν π)}$

HereJ_ν(z)are the Bessel function of the first kind:

$J_{ν} (z) = \frac{{(z / 2)}^{ν}}{\sqrt{π} Γ (ν + 1 / 2)} \int_{0}^{π} \cos (z \cos (t)) \sin {(t)}^{2 ν} d t$

Tips

Calling贝斯lyfor a number that is not a symbolic object invokes the MATLAB^®贝斯lyfunction.
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,贝斯ly(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