10.4

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.4#Ex1	${\displaystyle{\displaystyle J_{-n}\left(z\right)=(-1)^{n}J_{n}\left(z\right)}}$ \BesselJ{-n}@{z} = (-1)^{n}\BesselJ{n}@{z}	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	BesselJ(- n, z) = (- 1)^(n)* BesselJ(n, z)	BesselJ[- n, z] == (- 1)^(n)* BesselJ[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex2	${\displaystyle{\displaystyle Y_{-n}\left(z\right)=(-1)^{n}Y_{n}\left(z\right)}}$ \BesselY{-n}@{z} = (-1)^{n}\BesselY{n}@{z}	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0,\Re((-(-n))+k+1)>0}}$	BesselY(- n, z) = (- 1)^(n)* BesselY(n, z)	BesselY[- n, z] == (- 1)^(n)* BesselY[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex3	${\displaystyle{\displaystyle{H^{(1)}_{-n}}\left(z\right)=(-1)^{n}{H^{(1)}_{n}}\left(z\right)}}$ \HankelH{1}{-n}@{z} = (-1)^{n}\HankelH{1}{n}@{z}		HankelH1(- n, z) = (- 1)^(n)* HankelH1(n, z)	HankelH1[- n, z] == (- 1)^(n)* HankelH1[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex4	${\displaystyle{\displaystyle{H^{(2)}_{-n}}\left(z\right)=(-1)^{n}{H^{(2)}_{n}}\left(z\right)}}$ \HankelH{2}{-n}@{z} = (-1)^{n}\HankelH{2}{n}@{z}		HankelH2(- n, z) = (- 1)^(n)* HankelH2(n, z)	HankelH2[- n, z] == (- 1)^(n)* HankelH2[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex5	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right)+iY_{\nu}\left(z\right)}}$ \HankelH{1}{\nu}@{z} = \BesselJ{\nu}@{z}+i\BesselY{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH1(nu, z) = BesselJ(nu, z)+ I*BesselY(nu, z)	HankelH1[\[Nu], z] == BesselJ[\[Nu], z]+ I*BesselY[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex6	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right)-iY_{\nu}\left(z\right)}}$ \HankelH{2}{\nu}@{z} = \BesselJ{\nu}@{z}-i\BesselY{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH2(nu, z) = BesselJ(nu, z)- I*BesselY(nu, z)	HankelH2[\[Nu], z] == BesselJ[\[Nu], z]- I*BesselY[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex7	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2}\left({H^{(1)}_{\nu}}\left(z\right)+{H^{(2)}_{\nu}}\left(z\right)\right)}}$ \BesselJ{\nu}@{z} = \frac{1}{2}\left(\HankelH{1}{\nu}@{z}+\HankelH{2}{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = (1)/(2)*(HankelH1(nu, z)+ HankelH2(nu, z))	BesselJ[\[Nu], z] == Divide[1,2]*(HankelH1[\[Nu], z]+ HankelH2[\[Nu], z])	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex8	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{2i}\left({H^{(1)}_{\nu}}\left(z\right)-{H^{(2)}_{\nu}}\left(z\right)\right)}}$ \BesselY{\nu}@{z} = \frac{1}{2i}\left(\HankelH{1}{\nu}@{z}-\HankelH{2}{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = (1)/(2*I)*(HankelH1(nu, z)- HankelH2(nu, z))	BesselY[\[Nu], z] == Divide[1,2*I]*(HankelH1[\[Nu], z]- HankelH2[\[Nu], z])	Successful	Successful	-	Successful [Tested: 70]
10.4.E5	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-Y_{\nu}\left(z\right)\cos\left(\nu\pi\right)\right)}}$ \BesselJ{\nu}@{z} = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-\BesselY{\nu}@{z}\cos@{\nu\pi}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(-\nu))+k+1)>0}}$	BesselJ(nu, z) = csc(nu*Pi)*(BesselY(- nu, z)- BesselY(nu, z)*cos(nu*Pi))	BesselJ[\[Nu], z] == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- BesselY[\[Nu], z]*Cos[\[Nu]*Pi])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4#Ex9	${\displaystyle{\displaystyle{H^{(1)}_{-\nu}}\left(z\right)=e^{\nu\pi i}{H^{(1)}_{\nu}}\left(z\right)}}$ \HankelH{1}{-\nu}@{z} = e^{\nu\pi i}\HankelH{1}{\nu}@{z}		HankelH1(- nu, z) = exp(nu*Pi*I)*HankelH1(nu, z)	HankelH1[- \[Nu], z] == Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z]	Successful	Failure	-	Successful [Tested: 70]
10.4#Ex10	${\displaystyle{\displaystyle{H^{(2)}_{-\nu}}\left(z\right)=e^{-\nu\pi i}{H^{(2)}_{\nu}}\left(z\right)}}$ \HankelH{2}{-\nu}@{z} = e^{-\nu\pi i}\HankelH{2}{\nu}@{z}		HankelH2(- nu, z) = exp(- nu*Pi*I)*HankelH2(nu, z)	HankelH2[- \[Nu], z] == Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z]	Successful	Failure	-	Successful [Tested: 70]
10.4.E7	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{\nu}\left(z\right)-J_{-\nu}\left(z\right)\right)}}$ \HankelH{1}{\nu}@{z} = i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH1(nu, z) = I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z))	HankelH1[\[Nu], z] == I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E7	${\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{\nu}\left(z\right)-J_{-\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-e^{-\nu\pi i}Y_{\nu}\left(z\right)\right)}}$ i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{-\nu\pi i}\BesselY{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(-\nu))+k+1)>0}}$	I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z)) = csc(nu*Pi)*(BesselY(- nu, z)- exp(- nu*Pi*I)*BesselY(nu, z))	I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z]) == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[- \[Nu]*Pi*I]*BesselY[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E8	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z\right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)}}$ \HankelH{2}{\nu}@{z} = i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	HankelH2(nu, z) = I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z))	HankelH2[\[Nu], z] == I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E8	${\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z\right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z\right)\right)}}$ i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{\nu\pi i}\BesselY{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0,\Re((-(-\nu))+k+1)>0}}$	I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z)) = csc(nu*Pi)*(BesselY(- nu, z)- exp(nu*Pi*I)*BesselY(nu, z))	I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z]) == Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselY[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data