Bessel Functions - 10.28 Wronskians and Cross-Products

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.28.E1	${\displaystyle{\displaystyle\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z\right)}}$ \Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselI(nu, z))*diff(BesselI(- nu, z), z)-diff(BesselI(nu, z), z)*(BesselI(- nu, z)) = BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)	Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, z] == BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.28.E1	${\displaystyle{\displaystyle I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z\right)=-2\sin\left(\nu\pi\right)/(\pi z)}}$ \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) = - 2*sin(nu*Pi)/(Pi*z)	BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.28.E2	${\displaystyle{\displaystyle\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z\right)}}$ \Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselK(nu, z))*diff(BesselI(nu, z), z)-diff(BesselK(nu, z), z)*(BesselI(nu, z)) = BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)	Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, z] == BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.28.E2	${\displaystyle{\displaystyle I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z\right)=1/z}}$ \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) = 1/z	BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] == 1/z	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]