# 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.
• 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:

BesselY threads elementwise over lists and matrices:

BesselY can be used with Interval and CenteredInterval objects:

### 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(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 :

Wolfram Research (1988), BesselY, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselY.html (updated 2022).

#### Text

Wolfram Research (1988), BesselY, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselY.html (updated 2022).

#### CMS

Wolfram Language. 1988. "BesselY." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/BesselY.html.

#### APA

Wolfram Language. (1988). BesselY. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselY.html

#### BibTeX

@misc{reference.wolfram_2024_bessely, author="Wolfram Research", title="{BesselY}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/BesselY.html}", note=[Accessed: 17-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_bessely, organization={Wolfram Research}, title={BesselY}, year={2022}, url={https://reference.wolfram.com/language/ref/BesselY.html}, note=[Accessed: 17-June-2024 ]}