Bessel Functions - 10.5 Wronskians and Cross-Products

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.5.E1	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),J_{-\nu}\left(z\right)\right\}=J_{\nu+1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}\left(z\right)J_{-\nu-1}\left(z\right)}}$ \Wronskian@{\BesselJ{\nu}@{z},\BesselJ{-\nu}@{z}} = \BesselJ{\nu+1}@{z}\BesselJ{-\nu}@{z}+\BesselJ{\nu}@{z}\BesselJ{-\nu-1}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu-1)+k+1)>0}}$	(BesselJ(nu, z))*diff(BesselJ(- nu, z), z)-diff(BesselJ(nu, z), z)*(BesselJ(- nu, z)) = BesselJ(nu + 1, z)*BesselJ(- nu, z)+ BesselJ(nu, z)*BesselJ(- nu - 1, z)	Wronskian[{BesselJ[\[Nu], z], BesselJ[- \[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]*BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]*BesselJ[- \[Nu]- 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E1	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}\left(z\right)J_{-\nu-1}\left(z\right)=-2\sin\left(\nu\pi\right)/(\pi z)}}$ \BesselJ{\nu+1}@{z}\BesselJ{-\nu}@{z}+\BesselJ{\nu}@{z}\BesselJ{-\nu-1}@{z} = -2\sin@{\nu\pi}/(\pi z)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu-1)+k+1)>0}}$	BesselJ(nu + 1, z)*BesselJ(- nu, z)+ BesselJ(nu, z)*BesselJ(- nu - 1, z) = - 2*sin(nu*Pi)/(Pi*z)	BesselJ[\[Nu]+ 1, z]*BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]*BesselJ[- \[Nu]- 1, z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E2	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),Y_{\nu}\left(z\right)\right\}=J_{\nu+1}\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}\left(z\right)Y_{\nu+1}\left(z\right)}}$ \Wronskian@{\BesselJ{\nu}@{z},\BesselY{\nu}@{z}} = \BesselJ{\nu+1}@{z}\BesselY{\nu}@{z}-\BesselJ{\nu}@{z}\BesselY{\nu+1}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(\nu+1))+k+1)>0}}$	(BesselJ(nu, z))*diff(BesselY(nu, z), z)-diff(BesselJ(nu, z), z)*(BesselY(nu, z)) = BesselJ(nu + 1, z)*BesselY(nu, z)- BesselJ(nu, z)*BesselY(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], BesselY[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]*BesselY[\[Nu], z]- BesselJ[\[Nu], z]*BesselY[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E2	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}\left(z\right)Y_{\nu+1}\left(z\right)=2/(\pi z)}}$ \BesselJ{\nu+1}@{z}\BesselY{\nu}@{z}-\BesselJ{\nu}@{z}\BesselY{\nu+1}@{z} = 2/(\pi z)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(\nu+1))+k+1)>0}}$	BesselJ(nu + 1, z)*BesselY(nu, z)- BesselJ(nu, z)*BesselY(nu + 1, z) = 2/(Pi*z)	BesselJ[\[Nu]+ 1, z]*BesselY[\[Nu], z]- BesselJ[\[Nu], z]*BesselY[\[Nu]+ 1, z] == 2/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E3	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{(1)}_{\nu+1}}\left(z\right)}}$ \Wronskian@{\BesselJ{\nu}@{z},\HankelH{1}{\nu}@{z}} = \BesselJ{\nu+1}@{z}\HankelH{1}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{1}{\nu+1}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselJ(nu, z))*diff(HankelH1(nu, z), z)-diff(BesselJ(nu, z), z)*(HankelH1(nu, z)) = BesselJ(nu + 1, z)*HankelH1(nu, z)- BesselJ(nu, z)*HankelH1(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], HankelH1[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]*HankelH1[\[Nu], z]- BesselJ[\[Nu], z]*HankelH1[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E3	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{(1)}_{\nu+1}}\left(z\right)=2i/(\pi z)}}$ \BesselJ{\nu+1}@{z}\HankelH{1}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{1}{\nu+1}@{z} = 2i/(\pi z)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselJ(nu + 1, z)*HankelH1(nu, z)- BesselJ(nu, z)*HankelH1(nu + 1, z) = 2*I/(Pi*z)	BesselJ[\[Nu]+ 1, z]*HankelH1[\[Nu], z]- BesselJ[\[Nu], z]*HankelH1[\[Nu]+ 1, z] == 2*I/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}=J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)}}$ \Wronskian@{\BesselJ{\nu}@{z},\HankelH{2}{\nu}@{z}} = \BesselJ{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{2}{\nu+1}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselJ(nu, z))*diff(HankelH2(nu, z), z)-diff(BesselJ(nu, z), z)*(HankelH2(nu, z)) = BesselJ(nu + 1, z)*HankelH2(nu, z)- BesselJ(nu, z)*HankelH2(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], HankelH2[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- BesselJ[\[Nu], z]*HankelH2[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E4	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-2i/(\pi z)}}$ \BesselJ{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{2}{\nu+1}@{z} = -2i/(\pi z)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselJ(nu + 1, z)*HankelH2(nu, z)- BesselJ(nu, z)*HankelH2(nu + 1, z) = - 2*I/(Pi*z)	BesselJ[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- BesselJ[\[Nu], z]*HankelH2[\[Nu]+ 1, z] == - 2*I/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E5	${\displaystyle{\displaystyle\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}_{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)}}$ \Wronskian@{\HankelH{1}{\nu}@{z},\HankelH{2}{\nu}@{z}} = \HankelH{1}{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\HankelH{1}{\nu}@{z}\HankelH{2}{\nu+1}@{z}		(HankelH1(nu, z))*diff(HankelH2(nu, z), z)-diff(HankelH1(nu, z), z)*(HankelH2(nu, z)) = HankelH1(nu + 1, z)*HankelH2(nu, z)- HankelH1(nu, z)*HankelH2(nu + 1, z)	Wronskian[{HankelH1[\[Nu], z], HankelH2[\[Nu], z]}, z] == HankelH1[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- HankelH1[\[Nu], z]*HankelH2[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E5	${\displaystyle{\displaystyle{H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}_{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4i/(\pi z)}}$ \HankelH{1}{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\HankelH{1}{\nu}@{z}\HankelH{2}{\nu+1}@{z} = -4i/(\pi z)		HankelH1(nu + 1, z)*HankelH2(nu, z)- HankelH1(nu, z)*HankelH2(nu + 1, z) = - 4*I/(Pi*z)	HankelH1[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- HankelH1[\[Nu], z]*HankelH2[\[Nu]+ 1, z] == - 4*I/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]