8.57 Lommel functions

8.570 Definitions of the Lommel functions sμν(z) and Sμν(z):

1.12 

sμ,v(z)=m=0(1)mzμ+1+2m[(μ+1)2v2][(μ+3)2v2][(μ+2m+1)2v2]???????????????????????????????[μ±v?is?not?a?negative?odd?integer]=zμ1m=0(1)m(z2)2m+2Γ(12μ12v+12)Γ(12μ+12v+12)Γ(12μ12v+m+32)Γ(12μ+12v+m+32)???????????????????????????????????????????????????????[μ±v?is?not?a?negative?odd?integer]

si1107_e  EH II 40(69), WA 377(2)

2.12 

Sμ,v(z)=sμ,v(z)+2μ1Γ(12μ12v+12)Γ(12μ+12v+12)×cos[12(μv)π]Jv(z)cos[12(μ+v)π]Jv(z)sinvπ       [μ±v?is?a?positive?odd?integer,v?is?an?odd?integer]

  EH II 40(71), WA 379(2)

=sμ,v(z)+2μ1Γ(12μ12v+12)Γ(12μ+12v+12)×{sin[12(μ

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