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

# Examples

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## Basic Examples(6)

Evaluate numerically:

Compute the 5 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:

LegendreQ can be used with Interval and CenteredInterval objects:

### Specific Values(5)

Legendre function for :

Legendre function for fixed :

Find a local maximum as a root of :

Compute the associated Legendre function of the second kind :

Different LegendreQ types give different symbolic forms:

### Visualization(3)

Plot the LegendreQ function for various degrees:

Plot the real part of :

Plot the imaginary part of :

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

### Function Properties(12) 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 :

LegendreQ is not an analytic function:

Nor is it meromorphic: is neither non-decreasing nor non-increasing in for positive integer :

For and noninteger , it is increasing: is not injective in for positive integer :

For and noninteger , it is injective: 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,): is neither convex nor concave for most values of :

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the  derivative:

### Integration(3)

Indefinite integral of :

Definite integral of the odd integrand over the interval centered at the origin is 0:

Definite integral of the even integrand over the interval centered at the origin:

This is twice the integral over half the interval:

### Series Expansions(4)

Taylor expansion for :

Plot the first three approximations for at :

General term in the series expansion of :

Taylor expansion for the associated Legendre function :

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 :

## 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 PöschlTeller 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 PöschlTeller potential:

An n-point Gaussian quadrature rule is based on the roots of the n 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: