LegendreQ

LegendreQ[n,z]

gives the Legendre function of the second kind TemplateBox[{n, z}, LegendreQ].

LegendreQ[n,m,z]

gives the associated Legendre function of the second kind TemplateBox[{n, m, z}, LegendreQ3].

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.
  • LegendreQ can be used with Interval and CenteredInterval objects. »

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  (42)

Numerical Evaluation  (6)

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:

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 LegendreQ function using MatrixFunction:

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, z}, LegendreQ]:

Plot the imaginary part of TemplateBox[{4, z}, LegendreQ]:

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

Function Properties  (12)

TemplateBox[{m, z}, LegendreQ] is defined for as long as is not a negative integer:

In the complex plane, it is defined for as long as is not a negative integer:

The range for Legendre functions of integer order:

A Legendre function of an odd order is even:

A 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]:

LegendreQ is not an analytic function:

Nor is it meromorphic:

TemplateBox[{m, x}, LegendreQ] is neither non-decreasing nor non-increasing in for positive integer :

For and noninteger , it is increasing:

TemplateBox[{m, x}, LegendreQ] is not injective in for positive integer :

For and noninteger , it is injective:

TemplateBox[{m, x}, LegendreQ] is surjective in for non-negative even :

It is not surjective for other values of :

LegendreQ is neither non-negative nor non-positive:

LegendreQ has both singularity and discontinuity in (-∞,-1] and [1,∞):

TemplateBox[{m, x}, LegendreQ] is neither convex nor concave for most values of :

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  (4)

Angular momentum eigenfunctions:

Solve a recurrence equation:

The Pöschl–Teller potential is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions.

Find quantum eigenfunctions for the modified Pöschl–Teller potential:

An n-point Gaussian quadrature rule is based on the roots of the n^(th) order Legendre polynomial. Compute the nodes and weights of an n-point Gaussian quadrature rule:

Use the n-point Gaussian quadrature rule to numerically evaluate an integral:

The Kronrod extension of a Gaussian quadrature rule adds n+1 points and reuses the n nodes from Gaussian quadrature, resulting in an integration rule with 2n+1 points. The additional n+1 nodes can be obtained as the roots of a polynomial constructed from the asymptotic expansion of the Legendre function of the second kind (the Stieltjes polynomial):

Compute the Gauss–Kronrod nodes and weights:

Use the (2n+1)-point Gauss–Kronrod rule to numerically evaluate an integral:

The difference between the results of the Gauss–Kronrod rule and the Gaussian rule can be used as an error estimate:

Compare the result of the Gauss–Kronrod rule with the result from NIntegrate:

Properties & Relations  (2)

LegendreQ can be expressed as a DifferenceRoot:

The generating function for LegendreQ:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_legendreq, author="Wolfram Research", title="{LegendreQ}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LegendreQ.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_legendreq, organization={Wolfram Research}, title={LegendreQ}, year={2022}, url={https://reference.wolfram.com/language/ref/LegendreQ.html}, note=[Accessed: 22-December-2024 ]}