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LegendreQ

LegendreQ
gives the Legendre function of the second kind .
LegendreQ
gives the associated Legendre function of the second kind .
  • 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 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.
  • LegendreQ can be evaluated to arbitrary numerical precision.
Compute the 5^(th) Legendre function of the second kind:
Plot:
Compute the 5^(th) Legendre function of the second kind:
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Click for copyable input
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Plot:
In[1]:=
Click for copyable input
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:
TraditionalForm formatting:
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:
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