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BesselY

gives the Bessel function of the second kind .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
satisfies the differential equation .
BesselY[n,z] has a branch cut discontinuity in the complex z plane running from to .
FullSimplify and FunctionExpand include transformation rules for BesselY.
For certain special arguments, BesselY automatically evaluates to exact values.
BesselY can be evaluated to arbitrary numerical precision.
BesselY automatically threads over lists.
BesselY can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (44)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments and parameters:

Evaluate BesselY efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix BesselY function using MatrixFunction:

Specific Values (4)

Value of BesselY for integers () orders at :

For half-integer indices, BesselY evaluates to elementary functions:

Limiting value at infinity:

The first three zeros of :

Find the first zero of using Solve:

Visualize the result:

Visualization (3)

Plot the BesselY function for integer orders ():

Plot the real and imaginary parts of the BesselY function for integer orders ():

Plot the real part of :

Plot the imaginary part of :

Function Properties (10)

is defined for all real values greater than 0:

Complex domain:

Approximate function range of :

Approximate function range of :

is not an analytic function:

BesselY is neither non-decreasing nor non-increasing:

BesselY is not injective:

BesselY is not surjective:

BesselY is neither non-negative nor non-positive:

has both singularity and discontinuity for z≤0:

BesselY is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the derivative:

Integration (3)

Indefinite integral of BesselY:

Integrate expressions involving BesselY:

Definite integral of BesselY over its real domain:

Series Expansions (5)

Taylor expansion for around :

Plot the first three approximations for around :

General term in the series expansion of BesselY:

Asymptotic approximation of BesselY:

Taylor expansion at a generic point:

BesselY can be applied to a power series:

Integral Transforms (3)

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

MellinTransform:

Function Identities and Simplifications (3)

Use FullSimplify to simplify Bessel functions:

Recurrence relation :

For integer and arbitrary fixed , $TemplateBox[{{-, n}, z}, BesselY]=(-1)^n TemplateBox[{n, z}, BesselY]$ :

Function Representations (4)

Integral representation of BesselY:

Represent using BesselJ and Sin for non-integer :

BesselY can be represented in terms of MeijerG:

BesselY can be represented as a DifferenceRoot:

Applications (2)

Solve the Bessel differential equation:

Solve a differential equation:

Solve the inhomogeneous Bessel differential equation:

Properties & Relations (3)

Use FullSimplify to simplify Bessel functions:

BesselY can be represented as a DifferentialRoot:

The exponential generating function for BesselY:

Possible Issues (1)

With numeric arguments, half-integer Bessel functions are not automatically evaluated:

For symbolic arguments they are:

This can lead to major inaccuracies in machine-precision evaluation:

Neat Examples (1)

Plot the Riemann surface of :

Plot the Riemann surface of :

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