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Mathematica > Mathematics and Algorithms > Mathematical Functions > Special Functions >

Built-in Mathematica Symbol

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

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.

For certain special arguments, LegendreQ automatically evaluates to exact values.

Compute the 5

Legendre function of the second kind:

Plot:

Compute the 5

Legendre function of the second kind:

Plot:

Compute the associated Legendre function of the second kind:

Evaluate for fractional orders:

Evaluate for complex arguments:

Evaluate to high precision:

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

LegendreQ threads element-wise over lists:

LegendreQ can be applied to a power series:

Different LegendreQ types give different symbolic forms:

Types 2 and 3 have different branch cut structures:

Angular momentum eigenfunctions:

Solve a recurrence equation:

Use FunctionExpand to expand LegendreQ of integer or half-integer parameters into simpler functions: