# BesselY

BesselY[n,z]

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.

# 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(43)

### Numerical Evaluation(5)

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:

BesselY threads elementwise over lists and matrices:

### 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 z0:

BesselY is neither convex nor concave:

### 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:

### Function Identities and Simplifications(3)

Use FullSimplify to simplify Bessel functions:

Recurrence relation :

For integer and arbitrary fixed , :

### 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(1)

Solve the 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: