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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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7.16 Combinations of associated Legendre functions, powers, and trigonometric functions

7.161

1. 

01xλ1(1x2)12μsin(ax)Pvμ(x)dx=π1/22μλ1Γ(λ+1)aΓ(1+λμv2)Γ(3+λμ+v2)×2F3(1+λ2,1+λ2;32,1+λμ+v2,3+λμ+v2;a24)

si145_e  ET II 314(7)

[Reλ>1,Reμ<1]

si146_e

2. 

01xλ1(1x2)12μcos(ax)Pvμ(x)dx=π1/22μλΓ(λ)Γ(1+λμ+v2)Γ(1+λμ+v2)×2F3(λ2,λ+12;12,1+λμv2,1+λμ+v2;a24)

si147_e  ET II 314(8)

[Reλ>0,Reμ<1]

3. 

0(x21)12μsin(ax)Pvμ(x)dx=2μπ1/2aμ12Γ(1212μ12v)

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