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  (37)

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 TemplateBox[{0, x}, BesselY]:

Find the first zero of TemplateBox[{0, x}, BesselY] 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 TemplateBox[{0, {x, +, {ⅈ,  , y}}}, BesselY]:

Plot the imaginary part of TemplateBox[{0, {x, +, {ⅈ,  , y}}}, BesselY]:

Function Properties  (4)

TemplateBox[{n, z}, BesselY] is defined for all real values greater than 0:

Complex domain:

Approximate function range of TemplateBox[{1, z}, BesselY]:

Approximate function range of TemplateBox[{2, z}, BesselY]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the ^(th) 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 TemplateBox[{0, x}, BesselY] around :

Plot the first three approximations for TemplateBox[{0, x}, BesselY] 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 z (TemplateBox[{{n, -, 1}, z}, BesselY]+TemplateBox[{{n, +, 1}, z}, BesselY])=2 n TemplateBox[{n, z}, BesselY]:

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  (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:

Introduced in 1988
 (1.0)
 |
Updated in 1999
 (4.0)
2000
 (4.1)
2002
 (4.2)