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

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7

Definite Integrals of Special Functions

7.1-7.2 Associated Legendre Functions

7.11 Associated Legendre functions

7.111

cosφ1Pv(x)dx=sinφPv1(cosφ)

si1_e  MO 90

7.112

1. 

11Pnm(x)Pkm(x)dx=0[nk]=22n+1(n+m)!(nm)![n=k]

si2_e  SM III 185, WH

2. 

11Qnm(x)Pnm(x)dx=(1)m1(1)n+k(n+m)!(kn)(k+n+1)(nm)!

si3_e  EH I 171(18)

3. 

11Pv(x)Pσ(x)dx=2πsinπ(σv)+4sin(πv)sin(πσ)[ψ(v+1)ψ(σ+1)]π2(σv)(σ+v+1)[σ+v+10]=π22(sinπv)2ψ(v+1)π2(v+12)[σ=v]

  EH I 170(7) ...

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