LegendreQ

LegendreQ[n,z]

gives the Legendre function of the second kind .

LegendreQ[n,m,z]

gives the associated Legendre function of the second kind .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For integers n and m, explicit formulas are generated.
  • The Legendre functions satisfy the differential equation .
  • LegendreQ[n,m,a,z] gives Legendre functions of type a. The default is type 1.
  • LegendreQ of types 1, 2 and 3 are defined in terms of LegendreP of these types, and have the same branch cut structure and properties described for LegendreP.
  • For certain special arguments, LegendreQ automatically evaluates to exact values.
  • LegendreQ can be evaluated to arbitrary numerical precision.
  • LegendreQ automatically threads over lists.

Examples

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Basic Examples  (6)

Evaluate numerically:

Compute the 5^(th) Legendre function of the second kind:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Scope  (34)

Numerical Evaluation  (5)

Evaluate numerically at fixed points:

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate LegendreQ efficiently at high precision:

LegendreQ threads elementwise over lists:

Specific Values  (5)

Legendre function for :

Legendre function for fixed :

Find a local maximum as a root of (dTemplateBox[{5, x}, LegendreP])/(dx)=0:

Compute the associated Legendre function of the second kind TemplateBox[{3, 1, x}, LegendreQ3]:

Different LegendreQ types give different symbolic forms:

Visualization  (3)

Plot the LegendreQ function for various degrees:

Plot the real part of TemplateBox[{4, {x, +, {ⅈ,  , y}}}, LegendreQ]:

Plot the imaginary part of TemplateBox[{4, {x, +, {ⅈ,  , y}}}, LegendreQ]:

Type 2 and 3 Legendre functions have different branch cut structures:

Function Properties  (5)

The domain of a Legendre function of an integer order:

The range for a Legendre function of an integer order:

Legendre function of an odd order is even:

Legendre function of an even order is odd:

Legendre function has the mirror property TemplateBox[{n, TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, LegendreQ]=TemplateBox[{TemplateBox[{n, z}, LegendreQ]}, Conjugate]:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of TemplateBox[{1, x}, LegendreQ]:

Definite integral of the odd integrand TemplateBox[{2, x}, LegendreQ] over the interval centered at the origin is 0:

Definite integral of the even integrand TemplateBox[{3, x}, LegendreQ] over the interval centered at the origin:

This is twice the integral over half the interval:

Series Expansions  (4)

Taylor expansion for TemplateBox[{n, x}, LegendreQ]:

Plot the first three approximations for TemplateBox[{6, x}, LegendreQ] at :

General term in the series expansion of TemplateBox[{n, x}, LegendreQ]:

Taylor expansion for the associated Legendre function TemplateBox[{n, m, x}, LegendreQ3]:

LegendreQ can be applied to a power series:

Function Identities and Simplifications  (2)

Expand LegendreQ of integer or half-integer parameters into simpler functions:

Recurrence relation:

Function Representations  (4)

LegendreQ can be expressed as a DifferentialRoot:

Associated Legendre function in terms of the angular spheroidal function:

Associated Legendre function in terms of Legendre function of type :

TraditionalForm formatting:

Generalizations & Extensions  (2)

Different LegendreQ types give different symbolic forms:

Types 2 and 3 have different branch cut structures:

Applications  (2)

Angular momentum eigenfunctions:

Solve a recurrence equation:

Properties & Relations  (2)

LegendreQ can be expressed as a DifferenceRoot:

The generating function for LegendreQ:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)