# 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 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:

### 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(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 :

### 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(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)