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:

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:

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:

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 :

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the derivative:

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:

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:

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

Recurrence relation:

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: