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

BUILT-IN MATHEMATICA SYMBOL

LegendreQ

gives the Legendre function of the second kind

gives the associated Legendre function of the second kind

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:

Out[1]=

Plot:

Out[1]=

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: