LegendreQ
✖
LegendreQ
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
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0j46t20j-e4txqj

Compute the 5 Legendre function of the second kind:

https://wolfram.com/xid/0j46t20j-z7vuc

Plot over a subset of the reals:

https://wolfram.com/xid/0j46t20j-dfly1j

Plot over a subset of the complexes:

https://wolfram.com/xid/0j46t20j-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0j46t20j-evkb53

Asymptotic expansion at Infinity:

https://wolfram.com/xid/0j46t20j-nuxaho

Scope (42)Survey of the scope of standard use cases
Numerical Evaluation (6)
Evaluate numerically at fixed points:

https://wolfram.com/xid/0j46t20j-4lg4g


https://wolfram.com/xid/0j46t20j-dddg44

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

https://wolfram.com/xid/0j46t20j-bf2cjp

Evaluate for complex arguments:

https://wolfram.com/xid/0j46t20j-k0gdc

Evaluate LegendreQ efficiently at high precision:

https://wolfram.com/xid/0j46t20j-di5gcr


https://wolfram.com/xid/0j46t20j-bq2c6r

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

https://wolfram.com/xid/0j46t20j-c243ww


https://wolfram.com/xid/0j46t20j-cmdnbi


https://wolfram.com/xid/0j46t20j-f7nkq

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0j46t20j-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0j46t20j-thgd2

Or compute the matrix LegendreQ function using MatrixFunction:

https://wolfram.com/xid/0j46t20j-o5jpo

Specific Values (5)

https://wolfram.com/xid/0j46t20j-o8wq3z


https://wolfram.com/xid/0j46t20j-5l8vz

Find a local maximum as a root of :

https://wolfram.com/xid/0j46t20j-f2hrld


https://wolfram.com/xid/0j46t20j-gs8pkg

Compute the associated Legendre function of the second kind :

https://wolfram.com/xid/0j46t20j-jnqke

Different LegendreQ types give different symbolic forms:

https://wolfram.com/xid/0j46t20j-fp3rez

Visualization (3)
Plot the LegendreQ function for various degrees:

https://wolfram.com/xid/0j46t20j-ecj8m7


https://wolfram.com/xid/0j46t20j-ouu484


https://wolfram.com/xid/0j46t20j-cx88fp

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

https://wolfram.com/xid/0j46t20j-hj4yxr


https://wolfram.com/xid/0j46t20j-mtwnb

Function Properties (12)
is defined for
as long as
is not a negative integer:

https://wolfram.com/xid/0j46t20j-cl7ele

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

https://wolfram.com/xid/0j46t20j-7l7vpa

The range for Legendre functions of integer order:

https://wolfram.com/xid/0j46t20j-evf2yr

A Legendre function of an odd order is even:

https://wolfram.com/xid/0j46t20j-dnla5q

A Legendre function of an even order is odd:

https://wolfram.com/xid/0j46t20j-bi1e9n

Legendre function has the mirror property :

https://wolfram.com/xid/0j46t20j-heoddu

LegendreQ is not an analytic function:

https://wolfram.com/xid/0j46t20j-n3c7p8


https://wolfram.com/xid/0j46t20j-b7ce5m

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

https://wolfram.com/xid/0j46t20j-nc2cyz

For and noninteger
, it is increasing:

https://wolfram.com/xid/0j46t20j-ivvxpg

is not injective in
for positive integer
:

https://wolfram.com/xid/0j46t20j-gi38d7

For and noninteger
, it is injective:

https://wolfram.com/xid/0j46t20j-lq1zv8


https://wolfram.com/xid/0j46t20j-f2ik3p

is surjective in
for non-negative even
:

https://wolfram.com/xid/0j46t20j-z1c2ib

It is not surjective for other values of :

https://wolfram.com/xid/0j46t20j-uesmne


https://wolfram.com/xid/0j46t20j-bqlrsq

LegendreQ is neither non-negative nor non-positive:

https://wolfram.com/xid/0j46t20j-84dui

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

https://wolfram.com/xid/0j46t20j-mdtl3h


https://wolfram.com/xid/0j46t20j-mn5jws

is neither convex nor concave for most values of
:

https://wolfram.com/xid/0j46t20j-v9tr8p

Differentiation (3)

https://wolfram.com/xid/0j46t20j-mmas49


https://wolfram.com/xid/0j46t20j-nfbe0l


https://wolfram.com/xid/0j46t20j-fxwmfc


https://wolfram.com/xid/0j46t20j-odmgl1

Integration (3)

https://wolfram.com/xid/0j46t20j-d4t0yr

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

https://wolfram.com/xid/0j46t20j-lda193

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

https://wolfram.com/xid/0j46t20j-kn7g40

This is twice the integral over half the interval:

https://wolfram.com/xid/0j46t20j-ozzcy

Series Expansions (4)

https://wolfram.com/xid/0j46t20j-ewr1h8

Plot the first three approximations for at
:

https://wolfram.com/xid/0j46t20j-binhar

General term in the series expansion of :

https://wolfram.com/xid/0j46t20j-dznx2j

Taylor expansion for the associated Legendre function :

https://wolfram.com/xid/0j46t20j-j07vyq

LegendreQ can be applied to a power series:

https://wolfram.com/xid/0j46t20j-dbhj5m

Function Identities and Simplifications (2)
Expand LegendreQ of integer or half-integer parameters into simpler functions:

https://wolfram.com/xid/0j46t20j-fxzps5


https://wolfram.com/xid/0j46t20j-b6o1m


https://wolfram.com/xid/0j46t20j-nfe4y

Function Representations (4)
LegendreQ can be expressed as a DifferentialRoot:

https://wolfram.com/xid/0j46t20j-p1ybv

Associated Legendre function in terms of the angular spheroidal function:

https://wolfram.com/xid/0j46t20j-h1ek7l

Associated Legendre function in terms of Legendre function of type :

https://wolfram.com/xid/0j46t20j-kocyuf

TraditionalForm formatting:

https://wolfram.com/xid/0j46t20j-c62siy


https://wolfram.com/xid/0j46t20j-mwvwat

Generalizations & Extensions (2)Generalized and extended use cases
Different LegendreQ types give different symbolic forms:

https://wolfram.com/xid/0j46t20j-bpeoto


https://wolfram.com/xid/0j46t20j-dom455


https://wolfram.com/xid/0j46t20j-dwnym

Types 2 and 3 have different branch cut structures:

https://wolfram.com/xid/0j46t20j-dum9su


https://wolfram.com/xid/0j46t20j-dgvzwe

Applications (4)Sample problems that can be solved with this function
Angular momentum eigenfunctions:

https://wolfram.com/xid/0j46t20j-bgo7kq


https://wolfram.com/xid/0j46t20j-fhmsyu

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:

https://wolfram.com/xid/0j46t20j-i60rqx

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:

https://wolfram.com/xid/0j46t20j-j7hxj


https://wolfram.com/xid/0j46t20j-dtpkml

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

https://wolfram.com/xid/0j46t20j-g3ghz4

https://wolfram.com/xid/0j46t20j-fioj0d

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):

https://wolfram.com/xid/0j46t20j-7q7tu

Compute the Gauss–Kronrod nodes and weights:

https://wolfram.com/xid/0j46t20j-cbb4rf


https://wolfram.com/xid/0j46t20j-k4y7n9

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

https://wolfram.com/xid/0j46t20j-l1mhk3

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

https://wolfram.com/xid/0j46t20j-qq5t7

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

https://wolfram.com/xid/0j46t20j-b0w4n5

Properties & Relations (2)Properties of the function, and connections to other functions
LegendreQ can be expressed as a DifferenceRoot:

https://wolfram.com/xid/0j46t20j-dp28g8


https://wolfram.com/xid/0j46t20j-ck79j

The generating function for LegendreQ:

https://wolfram.com/xid/0j46t20j-pz93yz

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).
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.
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
Wolfram Language. (1988). LegendreQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LegendreQ.html
BibTeX
@misc{reference.wolfram_2025_legendreq, author="Wolfram Research", title="{LegendreQ}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LegendreQ.html}", note=[Accessed: 17-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_legendreq, organization={Wolfram Research}, title={LegendreQ}, year={2022}, url={https://reference.wolfram.com/language/ref/LegendreQ.html}, note=[Accessed: 17-May-2025
]}